Chapter 5 - Pearson Education

Chapter 5
Trigonometric Identities
Section 5.1
1. -2.6
Fundamental Identities
19. The quadrants are given so that one can
determine which sign (+ or -) sin θ will
1
, the sign of sin θ
csc θ
will be the same as csc θ .
take. Since sin θ =
2. -0.65
3. 0.625
4. -0.75
5.
2
3
6.
1
.
5
7.
8.
20. The range of cosine is [−1, 1], thus there is no
number or angle whose cosine is 3.
21. -sin x
22. odd
23. cos x
7
4
11
6
5 26
9. 26
24. even
25. - tan x
26. odd
27. This is the graph of f ( x) = sec x. It is
symmetric about the y-axis. Moreover, since
1
1
f (-x) = sec (-x) =
=
cos (-x ) cos x
10. -
3 10
10
11. -
2 5
5
28.
f (-x) = - f ( x)
33
12. 6
29.
f (-x) = - f ( x)
15
5
30.
f (-x) = - f ( x)
13. -
77
11
105
15. 11
14. -
= sec x = f ( x) ,
f (-x) = f ( x).
31. cos  = cot  = -
5
2 5
; tan  = ;
3
5
5
3 5
3
;sec  = ; csc  =
2
5
2
6
2 6
;
; tan  = 2 6; cot  =
12
5
16. -
3 5
7
32. sin  =
17. -
4
9
18. -
5
8
5 6
12
17
4 17
; cos  =
;
33. cot  = -4; sec  =
4
17
sec  = 5;csc  =
sin  = -
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17
; csc  = - 17
17
53
54
Chapter 5 Trigonometric Identities
2 21
21
2
;
; tan  =
34. sin  = - ; cos  = 21
5
5
cot  =
54. cot x =
 1- sin 2 x
sin x
55. tan x =  sec2 x -1
56. cot x =  csc2 x -1
3
4
3
36. cos  = - ; tan  = ; cot  = ;
5
3
4
5
5
sec  = - ; csc  = 3
4
57. csc x =
3
7
37. cos  = ;sin  = 4
4
7
3 7
4 7
tan  = ;cot  = ;csc  = 3
7
7
15
15
; tan  = - 15;cot  = ;
4
15
sec  = -4;csc  =
2 2 x + 4
x
53. sin x =  1- cos 2 x
21
5 21
;sec  = 2
21
3
5
4
35. tan = ;sec  = ;cos  = ;
4
4
5
3
5
sin  = ; csc  =
5
3
38. sin  =
52. tan  =
4 15
15
58. sec x =
 1- cos 2 x
1- cos 2 x
 1- sin 2 x
1- sin 2 x
59. cos 
60. sin 
61. 1
62. 1
63. cot 
39. B
64. tan 
40. D
65. cos 2 
41. E
42. C
66. csc 2 
43. A
67. sec  - cos 
44. C
68. tan 2 
45. A
69. - cot  + 1
46. E
70. tan  + 1
47. D
71. sin 2  cos 2 
48. B
2
2
49. It is incorrect to state 1 + cot = csc .
Cotangent and cosecant are functions of some
variable such as θ , x, or t. An acceptable
2
2
statement would be 1 + cot θ = csc θ .
50. In general, it is false that
2
2
x 2 + y 2 = x + y.
Stating sin θ + cos θ = 1 implies
sin θ + cos θ = 1 is a false statement.
 2 x +1
51. sin  =
x +1
72. sin 2  cos 2 
73. tan  sin 
74. cos 2  csc  or cot  cos 
75. cot  - tan 
76. tan  - cot 
77. cos 2 
78. - sin 2 
79. tan 2 
Copyright © 2013 Pearson Education, Inc.
Section 5.2 Verifying Trigonometric Identities
80. tan 4 
11. -
2
81. sec 
12. -
82. - tan 2 
83. - sec 
2 cos x
sin 2 x
2
2
cos 
or - 2 cot x csc x
or - 2sec2 
13. (sin  + 1)(sin  - 1)
84. - sin 
14. (sec  + 1)(sec  - 1)
25 6 - 60 -25 6 - 60
;
85.
12
12
2 2 + 8 -2 2 + 8
;
86.
9
9
87. y = - sin (2 x)
15. 4sin x
16. 4
17.
(2sin x +1)(sin x +1)
18.
(4 tan  - 3)( tan  +1)
89. y = cos (4 x)
19.
(cos2 x +1)
90. It is the same function.
20. csc2 x cot 2 x + 2
88. It is the negative of sin (2 x).
(
91. (a) y = - sin (4 x)
(b) y = cos (2 x)
(c)
y = 5sin (3x)
1
or csc  sec 
sin  cos 
1
2.
or csc x sec x
sin x cos x
3. 1 + sec x
4. 1 + cot 
5. 1
6. 1
7. 1 - 2 sin  cos 
8. sec 2 x + csc 2 x
9. 2 + 2 sin t
21.
(sin x - cos x)(1 + sin x cos x)
22.
(sin  + cos  )(1- sin  cos  )
24. cos 
identity
not an identity
not an identity
identity
25. 1
26. 1
Section 5.2 Verifying Trigonometric
Identities
1.
)
23. sin 
92. Answers will vary.
93.
94.
95.
96.
2
27. tan 2 
28. sec 2 
29. tan 2 x
30. cot 2 t
31. sec 2 x
32. csc 2 
33. cos 2 x
34. sin 2 x
cos θ
cot θ
cos θ sin θ
= sin θ =
⋅
= cos θ
35.
1
csc θ
sin θ 1
sin θ
10. sec 2 
Copyright © 2013 Pearson Education, Inc.
55
56
Chapter 5 Trigonometric Identities
sin α
tan α
cos
α = sin α ⋅ cos α = sin α
36.
=
1
sec α
cos α
cos α
37.
38.
1- sin 2 β cos 2 β
=
= cos β
cos β
cos β
æ sin 2 
ö÷
39. cos 2  (tan 2  + 1) = cos 2  ççç
+ 1÷÷÷
çè cos 2 
÷ø
æ sin 2  cos 2  ö÷
ç
÷÷
= cos 2  çç
+
çè cos 2  cos 2  ÷÷ø
æ sin 2  + cos 2  ÷ö
ç
÷÷
= cos 2  çç
÷÷
çè
cos 2 
ø
æ 1 ÷ö
÷=1
= cos 2  çç
çè cos 2  ÷÷ø
40. sin 2 β (1 + cot 2 β ) = sin 2 β csc2 β
1
= sin 2 β ⋅
=1
sin 2 β
cos  sin 
41. cot  + tan  =
+
sin  cos 
cos 2 
sin 2 
=
+
sin  cos  sin  cos 
cos 2  + sin 2 
1
=
cos  sin 
cos  sin 
1
1
=
⋅
= sec  csc 
cos  sin 
=
42. sin 2 α + tan 2 α + cos 2 α
)
= sin 2 α + cos 2 α + tan 2 α
= 1 + tan 2 α = sec2 α
43. Working with the left side, we have
cos α sin α
cos α sin α
+
=
+
1
1
sec α csc α
cos α sin α
2
sec 2 α - tan 2 α = 1.
cos α sin α
+
= 1 = sec 2 α - tan 2 α ,
sec α csc α
the statement has been verified.
Since
44.
tan 2 α + 1 sec2 α
=
= sec α
sec α
sec α
(
Working with the right side, we have
2
= cos α + sin α = 1
sin 2 θ 1- cos 2 θ
1
cos 2 θ
=
=
cos θ
cos θ
cos θ
cos θ
= sec θ - cos θ
45. sin 4 θ - cos 4 θ
(
)(
)
= 1⋅ (sin 2 θ - cos 2 θ ) = sin 2 θ - cos 2 θ
= sin 2 θ - (1- sin 2 θ ) = 2sin 2 θ -1
= sin 2 θ + cos 2 θ sin 2 θ - cos 2 θ
46. Simplify the left side.
(
)
sec 4 x - sec2 x = sec 2 x sec 2 x -1
= sec 2 x tan 2 x = tan 2 x sec2 x
Simplify the right side.
(
)
tan 4 x + tan 2 x = tan 2 x tan 2 x + 1
= tan 2 x sec 2 x
sec 4 x - sec 2 x = tan 2 x sec 2 x = tan 4 x + tan 2 x
Thus, the statement has been verified.
47. Work with the left side.
1- cos x (1- cos x)(1- cos x)
=
1 + cos x (1 + cos x)(1- cos x)
=
=
1- 2 cos x + cos2 x
1- cos2 x
1- 2 cos x + cos2 x
sin 2 x
Work with the right side.
2
æ cos x
æ cos x -1ö÷2
1 ö÷
(cot x - csc x)2 = ççç
÷÷ = ççç
÷
è sin x sin x ø
è sin x ø÷
=
cos 2 x - 2 cos x + 1
sin 2 x
1- cos x cos 2 x - 2 cos x + 1
2
=
= (cot x - csc x )
1 + cos x
sin 2 x
Thus, the statement has been verified.
Copyright © 2013 Pearson Education, Inc.
Section 5.2 Verifying Trigonometric Identities
2
(sec  - tan  ) + 1
2
48. (sec α - tan α )
50.
= sec 2 α - 2sec α tan α + tan 2 α
1 sin α sin 2 α
=
- 2⋅
⋅
+
cos α cos α cos 2 α
cos 2 α
=
1- 2sin α + sin 2 α
2
cos α
sec  csc  - tan  csc 
=
1
2
=
(1- sin α )
=
2
1- sin α
57
sec2  - 2sec  tan  + tan 2  + 1
csc  (sec  - tan  )
(
)
sec2  - 2sec  tan  + tan 2  + 1
csc  (sec  - tan  )
2
=
(1- sin α )
1- sin α
=
(1- sin α )(1 + sin α ) 1 + sin α
=
49. Work with the left side.
cos θ + 1 cos θ + 1
cos θ + 1
=
=
2
2
1
tan θ
sec θ -1
-1
cos 2 θ
=
=
=
(cos θ + 1) cos 2 θ
æ 1
ö 2
ççç 2 -1÷÷÷ cos θ
è cos θ
ø
cos 2 θ (cos θ + 1)
51.
1- cos 2 θ
cos 2 θ (cos θ + 1)
(1 + cos θ )(1- cos θ )
=
cos 2 θ
1- cos θ
Now work with the right side.
cos θ
cos θ
cos θ cos θ
=
=
⋅
1
1
sec θ -1
-1
-1 cos θ
cos θ
cos θ
=
2sec 2  - 2sec  tan 
csc  (sec  - tan  )
=
2sec  (sec  - tan  )
csc  (sec  - tan  )
=
2sec 
sin 
= 2⋅
= 2 tan 
csc 
cos 
1
1
+
1- sin θ 1 + sin θ
1 + sin θ
1- sin θ
=
+
(1 + sin θ )(1- sin θ ) (1 + sin θ )(1- sin θ )
=
=
2
cos θ
1- cos θ
cos θ + 1
cos 2 θ
cos θ
=
=
2
1- cos θ sec θ -1
tan θ
Thus, the statement has been verified.
=
sec 2  - 2sec  tan  + sec2 
csc  (sec  - tan  )
52.
(1 + sin θ ) + (1- sin θ ) 1 + sin θ + 1- sin θ
=
(1 + sin θ )(1- sin θ ) (1 + sin θ )(1- sin θ )
2
2
1- sin θ
=
2
cos 2 θ
= 2sec 2 θ
1
1
sec  + tan 
=
⋅
sec  - tan  sec  - tan  sec  + tan 
sec  + tan 
=
sec2  - tan 2 
=
sec  + tan 
1
-
2
sin 
=
sec  + tan 
1- sin 2 
cos 2  cos 2 
cos 2 
sec  + tan  sec  + tan 
=
=
1
cos 2 
cos 2 
= sec  + tan 
Copyright © 2013 Pearson Education, Inc.
58
Chapter 5 Trigonometric Identities
53.
cos α
cos α
+1
+1
cot α + 1 sin α
sin α
=
= sin α
⋅
cot α -1 cos α -1 cos α -1 sin α
sin α
sin α
cos α + sin α
=
cos α - sin α
1
cos α + sin α cos α
=
⋅
1
cos α - sin α
cos α
cos α sin α
+
1 + tan α
= cos α cos α =
cos α sin α
1- tan α
cos α cos α
58.
55.
cos θ
=
sin θ cot θ
=
(1 + cos θ ) cos θ
cos θ
=
2
sin θ (1 + cos θ ) sin 2 θ
=
1 cos θ
⋅
= csc θ cot θ
sin θ sin θ
)
(
(sin 2 α + cos2 α )(sin 2 α - cos2 α )
sin 2 α - cos 2 α
= sin 2 α + cos 2 α = 1
59. Simplify the right side
1
tan t tan t - cot t
t
tan
=
tan t + cot t tan t + 1
tan t
1
tan t tan t ⋅ tan t
=
1 tan t
tan t +
tan t
=
tan 2 t + 1
=
tan 2 t -1
sec 2 t
cos 2 t
cos 2 t
1
-1
2
cot 2 t -1 sin 2 t
sin 2 t
=
= sin t
⋅
1 + cot 2 t
cos 2 t
cos 2 t sin 2 t
1+
1+ 2
2
sin t
sin t
=
cos 2 t - sin 2 t
sin 2 t + cos2 t
=
cos 2 t - sin 2 t
1
(
)
= cos 2 t - sin 2 t = 1- sin 2 t - sin 2 t
= 1- 2sin 2 t
sin 2 α sec2 α + sin 2 α csc2 α
)
56. sin 2 θ 1 + cot 2 θ -1 = sin 2 θ csc 2 θ -1
æ 1 ö÷
= sin 2 θ çç
÷ -1
çè sin 2 θ ÷÷ø
61.
= sin 2 α ⋅
=
sin 2 α
cos 2 α
= 1 -1 = 0
57.
tan 2 t -1
60.
cos θ
cos θ
=
=1
cos θ
cos θ
sin θ ⋅
sin θ
(
sin 2 α - cos 2 α
=
54.
1
cos θ
+
csc θ + cot θ
sin
θ
sin θ
=
sin θ
tan θ + sin θ
+ sin θ
cos θ
1 + cos θ
sin θ cos θ
sin θ
=
⋅
æ 1
ö sin θ cos θ
sin θ çç
+ 1÷÷÷
çè cos θ
ø
sin 4 α - cos 4 α
sec4 θ - tan 4 θ
sec2 θ + tan 2 θ
sec2 θ + tan 2 θ )(sec2 θ - tan 2 θ )
(
=
sec2 θ + tan 2 θ
= sec2 θ - tan 2 θ
Copyright © 2013 Pearson Education, Inc.
1
2
cos α
+ sin 2 α ⋅
1
sin 2 α
+ 1 = tan 2 α + 1 = sec 2 α
Section 5.2 Verifying Trigonometric Identities
62. Work with the left side.
2
2
(
2
2
tan α sin α = tan α 1- cos α
65.
)
1 + cos x 1- cos x
1- cos x 1 + cos x
= tan 2 α - tan 2 α cos 2 α
=
(1 + cos x)2
(1- cos x)2
(1 + cos x)(1- cos x) (1 + cos x)(1- cos x)
=
1 + 2 cos x + cos 2 x
1- 2 cos x + cos 2 x
(1 + cos x)(1- cos x) (1 + cos x)(1- cos x)
tan 2 α + cos 2 α -1 = tan 2 α - sin 2 α
=
= tan 2 α + cos 2 α -1
Thus, the statement has been verified.
1 + 2 cos x + cos 2 x -1 + 2 cos x - cos 2 x
(1 + cos x)(1- cos x)
=
2
2
= tan α - sin α
Now work with the right side.
(
tan 2 α + cos 2 α -1 = tan 2 α - 1- cos 2 α
)
= tan 2 α - sin 2 α
63.
59
tan x
sin x
+
1 + cos x 1- cos x
tan x (1- cos x)
sin x (1 + cos x)
=
+
(1 + cos x)(1- cos x) (1 + cos x)(1- cos x)
=
=
tan x (1- cos x) + sin x (1 + cos x)
tan x - sin x + sin x + sin x cos x
2
1- cos x
tan x + sin x cos x
tan x sin x cos x
=
=
+
2
sin x
sin 2 x
sin 2 x
1
cos x sin x
1
= tan x ⋅
+
=
⋅
+ cot x
2
sin x sin x cos x sin 2 x
1
1
=
⋅
+ cot x = sec x csc x + cot x
cos x sin x
66.
=
sin θ (1 + cos θ ) - sin θ cos θ (1- cos θ )
(1 + cos θ )(1- cos θ )
=
sin θ + sin θ cos θ - sin θ cos θ + sin θ cos 2 θ
=
sin θ + sin θ cos 2 θ
=
1- cos 2 θ
sin 2 θ
=
1 + cos 2 θ
sin θ
1
1 + cos 2 θ = csc θ 1 + cos 2 θ
sin θ
(
)
(
)
=
4 cos x
2
= 4⋅
cos x 1
⋅
sin x sin x
1 + sin θ 1- sin θ
1- sin θ 1 + sin θ
(1 + sin θ )2 - (1- sin θ )2
(1- sin θ )(1 + sin θ )
1 + 2sin θ + sin 2 θ ) - (1- 2sin θ + sin 2 θ )
(
=
1- sin 2 θ
4sin θ
sin θ
1
=
=4
⋅
= 4 tan θ sec θ
2
cos
cos
θ
θ
cos θ
67. Simplify the right side
sec2 θ - 2sec θ tan θ + tan 2 θ
=
64.
sin θ
sin θ cos θ
1- cos θ 1 + cos θ
sin θ (1 + cos θ )
sin θ cos θ (1- cos θ )
=
(1 + cos θ )(1- cos θ ) (1 + cos θ )(1- cos θ )
2
1- cos x sin x
= 4 cot x csc x
=
(1 + cos x )(1- cos x)
4 cos x
=
1
cos2 θ
- 2⋅
1 sin θ sin 2 θ
⋅
+
cos θ cos θ cos2 θ
1- 2sin θ + sin 2 θ
cos 2 θ
2
=
(1- sin θ )
1- sin 2 θ
2
=
(1- sin θ )
1- sin θ
=
(1 + sin θ )(1- sin θ ) 1 + sin θ
68. Simplify the right side.
sin θ
cos θ
+
1- cot θ 1- tan θ
sin θ
cos θ
=
+
cos θ
sin θ
11sin θ
cos θ
sin θ
sin θ
cos θ cos θ
=
⋅
+
⋅
cos θ sin θ
sin θ cos θ
11sin θ
cos θ
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60
Chapter 5 Trigonometric Identities
=
sin 2 θ
cos 2 θ
+
sin θ - cos θ cos θ - sin θ
Working with the left side, we have
=
sin 2 θ
cos 2 θ
+
sin θ - cos θ -(sin θ - cos θ )
= 1 + sin x + cos x + sin x + sin 2 x + sin x cos x
2
(1 + sin x + cos x )2
2
2
+ cos x + sin x cos x + cos 2 x
= 2 + 2sin x + 2 cos x + 2sin x cos x
2
sin θ
cos θ
sin θ - cos θ
=
sin θ - cos θ sin θ - cos θ
sin θ - cos θ
(sin θ + cos θ )(sin θ - cos θ )
=
= sin θ + cos θ
sin θ - cos θ
=
= 2 (1 + sin x )(1 + cos x )
Thus, the statement has been verified.
71.
69.
-1
-1
+
tan α - sec α tan α + sec α
- tan α - sec α
=
( tan α + sec α )( tan α - sec α )
= 2sin 2 α - sin 4 α
- tan α + sec α
+
( tan α + sec α )( tan α - sec α )
=
=
=
- tan α - sec α - tan α + sec α
( tan α + sec α )( tan α - sec α )
-2 tan α
2
2
tan α - sec α
=
-2 tan α
2
(
)
72. (sec  + csc  )(cos  - sin  )
æ 1
1 ö÷
= çç
+
cos  - sin  )
çè cos  sin  ø÷÷(
cos  sin  cos  sin 
=
+
cos  cos  sin  sin 
= 1- tan  + cot  -1 = cot  - tan 
tan α - tan 2 α + 1
-2 tan α
2
tan α - tan α -1
2
(1- cos2 α )(1+ cos2 α ) = sin 2 α (1+ cos2 α )
= sin 2 α (2 - sin 2 α )
=
-2 tan α
= 2 tan α
-1
70. Working with the right side, we have
2 (1 + sin x)(1 + cos x)
= 2 (1 + sin x + cos x + sin x cos x)
= 2 + 2sin x + 2 cos x + 2sin x cos x
73. Work with the left side:
1- cos x 1- cos x 1- cos x 1- 2 cos x + cos 2 x 1- 2 cos x + cos 2 x
=
⋅
=
=
1 + cos x 1 + cos x 1- cos x
1- cos 2 x
sin 2 x
Work with the right side:
1
2 cos x cos 2 x 1- 2 cos x + cos 2 x
csc 2 x - 2 csc x cot x + cot 2 x =
+
=
sin 2 x sin 2 x sin 2 x
sin 2 x
1- cos x 1- 2 cos x + cos 2 x
Since
=
= csc2 x - 2 csc x cot x + cot 2 x, the statement has been verified.
2
1 + cos x
sin x
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Section 5.2 Verifying Trigonometric Identities
61
74. Work with the left side:
1- cos θ 1- cos θ 1- cos θ 1- 2 cos θ + cos 2 θ 1- 2 cos θ + cos 2 θ
=
⋅
=
=
1 + cos θ 1 + cos θ 1- cos θ
1- cos 2 θ
sin 2 θ
Work with the right side:
(
)
2 csc 2 θ - 2 csc x cot θ -1 = 2 csc 2 θ - 2 csc θ cot θ - csc2 θ - cot 2 θ = csc2 θ - 2 csc θ cot θ + cot 2 θ
=
Since
1
sin 2 θ
-
2 cos θ
sin 2 θ
+
cos 2 θ
sin 2 θ
=
1- 2 cos θ + cos 2 θ
sin 2 θ
1- cos θ 1- 2 cos θ + cos 2 θ
=
= 2 csc 2 θ - 2 csc x cot θ -1, the statement has been verified.
2
1 + cos θ
sin θ
2
2
(
) (
75. (2sin x + cos x ) + (2 cos x - sin x ) = 4sin 2 x + 4sin x cos x + cos 2 x + 4 cos 2 x - 4sin x cos x + sin 2 x
(
) (
)
)
= 4 sin 2 x + cos 2 x + cos 2 x + sin 2 x = 4 + 1 = 5
76. sin 2 x (1 + cot x) + cos 2 x (1- tan x ) + cot 2 x = sin 2 x + sin 2 x cot x + cos 2 x - cos2 tan x + cot 2 x
æ cos x ö÷
æ sin x ö÷
= sin 2 x + cos 2 x + sin 2 x çç
- cos2 çç
+ cot 2 x
çè sin x ø÷÷
çè cos x ø÷÷
(
)
= 1 + sin x cos x - sin x cos x + cot 2 x = 1 + cot 2 x = csc2 x
77.
sec x - cos x + csc x - sin x - sin x tan x =
æ sin x ö÷
1
1
- cos x +
- sin x - sin x çç
çè cos x ÷÷ø
cos x
sin x
æ 1
ö æ 1
ö sin 2 x
= çç
- cos x÷÷÷ + çç
- sin x÷÷÷ çè cos x
ø çè sin x
ø cos x
1- cos 2 x 1- sin 2 x sin 2 x
+
cos x
sin x
cos x
æ1- cos 2 x sin 2 x ö÷ 1- sin 2 x 1- cos 2 x - sin 2 x cos 2 x
ç
÷÷ +
= çç
=
+
cos x ÷ø÷
sin x
cos x
sin x
çè cos x
=
=
(
(
1- cos 2 x + sin 2 x
cos x
) + cos2 x = 1-1 + cos x ⋅ cos x = cos x cot x
sin x
cos x
sin x
)
78. sin3 θ + cos3 θ = (sin θ + cos θ ) sin 2 θ - sin θ cos θ + cos2 θ = (cos θ + sin θ )(1- sin θ cos θ )
79.
(sec + tan  )(1- sin  ) = cos 
80.
(csc + cot  )(sec -1) = tan 
81.
cos  + 1
= cot 
sin  + tan 
82.
83.
84.
85.
86.
tan  sin  + cos  = sec 
identity
identity
not an identity
not an identity
87. Show that sin (csc t ) = 1 is not an identity.
We need to find only one value for which the
statement is false. Let t = 2. Use a calculator to
find that sin (csc 2) ≈ 0.891094, which is not
equal to 1. sin (csc t ) = 1 does not hold true
for all real numbers t. Thus, it is not an
identity.
Copyright © 2013 Pearson Education, Inc.
62
Chapter 5 Trigonometric Identities
88. Show that cos 2 t = cos t is not an identity.
π
π 1
Let t = . We have cos = and
3 2
3
æ 1 ö2
1 1
= çç ÷÷÷ =
= .
cos
ç
è 2ø
3
4 2
(
93. (a) I = k cos 2 θ = k 1- sin 2 θ
(b) For θ = 2π n for all integers n,
cos 2 θ = 1, its maximum value and I
attains a maximum value of k.
2π
Now, let t =
cos 2
1
2π
. We have, cos t = - and
2
3
94. (a) P = 16k cos 2 (2 t )
(b) P = 16k éê1- sin 2 (2 t )ùú
ë
û
2
æ 1ö
2π
1 1
= çç- ÷÷÷ =
= .
ç
è 2ø
3
4 2
)
95. (a) The sum of L and C equals 3.
cos 2 t = cos t does not hold true for all real
numbers t. Thus, it is not an identity.
89. Show that csc t = 1 + cot 2 t is not an
identity.
Let t =
π
4
1 + cot 2
. We have csc
π
4
π
4
= 2 and
= 1 + 12 = 1 + 1 = 2. But let
æ πö
π
t = - . We have csc çç- ÷÷÷ = - 2 and
çè 4 ø
4
(b) Let
Y1 = L (t ) , Y2 = C (t ) , and Y3 = E (t )
Y3 = 3 for all inputs.
æ πö
2
1 + cot 2 çç- ÷÷÷ = 1 + (-1) = 1 + 1 = 2.
çè 4 ø
csc t = 1 + cot 2 t does not hold true for all
real numbers t. Thus, it is not an identity.
90. Show that cos t = 1- sin 2 t is not an
identity.
π
π 1
Let t = . We have cos = and
3 2
3
1- sin 2
= 3cos 2 (6, 000, 000t ) + 3sin 2 (6, 000, 000t )
æ 3 ö÷2
3
1 1
= 1- ççç ÷÷ = 1- =
= .
çè 2 ÷ø
3
4
4 2
= 3 éêcos 2 (6, 000, 000t ) + sin 2 (6, 000, 000t )ùú
ë
û
= 3 ⋅1 = 3
π
But let t =
1- sin 2
(c) E (t ) = L (t ) + C (t )
2π
2π
1
. We have cos
= - and
3
3
2
æ 3 ö2
2π
3
= 1- ççç ÷÷÷ = 1çè 2 ÷ø
3
4
1 1
=
=
4 2
Section 5.3 Sum and Difference Identities
for Cosine
1. F
2. A
3. E
cos t = 1- sin 2 t does not hold true for all
real numbers t. Thus, it is not an identity.
91. a true statement when sin x £ 0
92. a true statement when cos x £ 0
4. B
5. E
6. C
7.
6- 2
4
Copyright © 2013 Pearson Education, Inc.
Section 5.3 Sum and Difference Identities for Cosine
8.
6+ 2
4
9.
2- 6
4
10.
2- 6
4
27. csc (-56 42 ¢)
28. cot (-843¢)
29. tan (-86.9814)
30. cos (-8.0142)
31. tan
11.
2- 6
4
32. cos
12.
6+ 2
4
34. tan
13.
6+ 2
4
36. tan
14.
2- 6
4
33. cos
35. csc
For exercises 37−42, other answers are possible.
37. 15
38. 20
15. 0
16. -1
17. The answer to exercise 15 is 0. Using a
calculator to evaluate
cos 40 cos 50 - sin 40 sin 50 also gives a
value of 0. (Make sure that your calculator is
in DEGREE mode.)
18. The answer to exercise 16 is −1. Using a
calculator to evaluate
7π
2π
7π
2π
also gives a
cos
cos
sin
- sin
9
9
9
9
value of −1. (Make sure that your calculator is
in RADIAN mode.)
19. cot 3°
20. cos 75
21. sin
22. cos
39.
140
3
40.
348
5
41. 20
42. 40
43. cos 
44. sin 
45. - cos 
46. - sin 
47. cos 
48. - sin 
5
12
49. - cos 

50. sin 
10
23. sec 7536¢
24. cos (-5214 ¢)
æ ö
25. cos çç- ÷÷÷
çè 8 ø
æ 2 ö
26. tan çç- ÷÷÷
çè 5 ø
16 56
;65 65
36 84
52. - ;
85 85
51.
53.
4-6 6 4 + 6 6
;
25
25
54.
-2 10 + 2 -2 10 - 2
;
9
9
Copyright © 2013 Pearson Education, Inc.
63
64
55.
Chapter 5 Trigonometric Identities
2 638 - 30 2 638 + 30
;
56
56
73.
62 - 70 62 + 70
;
24
24
57. true
56.
- 6- 2
4
74. (a)
58. false
(b)
59. false
2- 6
4
- 6- 2
4
60. true
75. (a) 3 cycles
61. true
(b) 163; –163; no
62. true
æ 20
ö
-1026t ÷÷÷
76. (a) Graph P = 0.04 cos çç
çè 4.9
ø
63. true
64. true
65. false
66. false
æπ
ö
π
π
67. cos çç + x÷÷÷ = cos cos x - sin sin x
çè 2
ø
2
2
= (0) cos x - (1) sin x = - sin x
68. sec (π - x) =
1
cos (π - x)
1
cos π cos x + sin π sin x
1
=
(-1) cos x + (0)sin x
=
=-
The pressure P is oscillating.
æ 2 r
ö
3
-10, 260÷÷÷
(b) Graph P = cos çç
ç
è
ø
r
4.9
1
= - sec x
cos x
69. cos 2 x = cos ( x + x)
= cos x cos x - sin x sin x
= cos 2 x - sin 2 x
70. From exercise 65, cos 2 x = cos 2 x - sin 2 x .
1 + cos 2 x - cos 2 x = 1 + cos 2 x - sin 2 x - cos 2 x
= 1- sin 2 x = cos 2 x
The pressure oscillates, and amplitude
decreases as r increases.
71. cos195 = cos (180 + 15)
= cos180 cos15- sin180 sin15
= (-1) cos15- (0) sin15
= - cos15- 0 = - cos15
72.
- 6- 2
4
(c)
a
cos (ct )
n
77. cos (90 +  ) = - sin 
78. cos ( 270- ) = - sin 
79. cos (180 +  ) = - cos 
Copyright © 2013 Pearson Education, Inc.
Section 5.4 Sum and Difference Identities for Sine and Tangent
80. cos (180- ) = - cos 
24. - 3
81. cos (270 +  ) = sin 
25. 1
26. 1
82. cot 
Section 5.4
Sum and Difference
Identities for Sine and
Tangent
1. C
2. A
3. E
4. F
27.
3 cos  - sin 
2
28.
3 cos  + sin 
2
29.
cos  - 3 sin 
2
30.
5. B
6. D
Answers will vary.
7.−8.
65
31.
9.
6+ 2
4
32.
10.
2- 6
4
33.
11. 2 - 3
34.
12. 2 + 3
13.
2+ 6
4
35.
14.
6- 2
4
36.
- 6- 2
15.
4
2 (cos  + sin  )
2
2 (sin x - cos x )
2
2 (sin  + cos  )
2
3 tan  + 1
3 - tan 
1 + tan x
1 - tan x
2 (cos x + sin x)
2
2 (cos x + sin x )
2
37. - cos 
38. tan 
39. - tan x
- 2- 6
16.
4
40. - sin x
17. 2 - 3
41. - tan x
18. 2 + 3
42. To follow the method of Example 2 to find
tan (270-θ ) , we need to use the tangent of a
19.
2
2
20. 1
21. -1
22. -1
difference formula:
tan 270- tan θ
tan (270- θ ) =
1 + tan 270 tan θ
However, tan 270º is undefined.
43. Answers will vary.
23. 0
Copyright © 2013 Pearson Education, Inc.
66
Chapter 5 Trigonometric Identities
44. If A, B, and C are angles of a triangle, then
A + B + C = 180. Therefore, we have
sin ( A + B + C ) = sin180 = 0.
45. (a)
63
65
(b)
63
16
-
(b) -
51.
63
65
- 6- 2
4
53. -2 + 3
54. -2 - 3
55. -2 + 3
63
16
56.
æ 3
ö
58. equivalent; sin çç +  ÷÷÷= - cos 
çè 2
ø
(c) quadrant II
-
(b)
36
77
36
85
æ
ö
59. equivalent; tan çç +  ÷÷÷= - cot 
çè 2
ø
æ
ö
60. equivalent; tan çç - ÷÷÷= cot 
çè 2
ø
61. Verify sin 2 x = 2 sin x cos x is an identity.
sin 2 x = sin ( x + x ) = sin x cos x + cos x sin x
(c) quadrant III
49. (a)
(b)
6- 2
4
æ
ö
57. equivalent; sin çç +  ÷÷÷ = cos 
çè 2
ø
77
36
48. (a)
6- 2
4
52.
77
85
(b) -
-8 6 - 3
25
6 +6
4
(c) quadrant III
(c) quadrant IV
47. (a)
50. (a)
(b)
(c) quadrant I
46. (a)
(c) quadrant II
4 2+ 5
9
= 2sin x cos x
- 5- 2
- 2- 5
or
2
2
62. Verify sin ( x + y ) + sin ( x - y ) = 2sin x cos y is an identity.
sin ( x + y ) + sin ( x - y ) = (sin x cos y + cos x sin y ) + (sin x cos y - cos x sin y ) = 2sin x cos y
æ 7π
ö
æ 2π
ö
63. Verify sin çç + x÷÷÷ - cos çç + x÷÷÷ = 0 is an identity.
çè 6
ç
ø
è 3
ø
æ 7π
ö÷
æ 2π
ö÷ æ 7π
ö æ
ö
7π
2π
2π
sin çç + x÷÷ - cos çç + x÷÷ = ççsin
cos x + cos
sin x÷÷÷ - ççcos
cos x - sin
sin x÷÷÷
çè 6
ç
ç
ç
ø
è 3
ø è
ø è
ø
6
6
3
3
æ 1
ö÷ æ 1
ö÷
3
3
sin x÷÷ - ççç- cos x sin x÷÷ = 0
= ççç- cos x ÷
2
2
èç 2
ø èç 2
ø÷
Copyright © 2013 Pearson Education, Inc.
Section 5.4 Sum and Difference Identities for Sine and Tangent
64. Verify tan ( x - y ) - tan ( y - x ) =
tan ( x - y ) - tan ( y - x ) =
2 ( tan x - tan y )
1 + tan x tan y
67
is an identity.
tan x - tan y
tan y - tan x
tan x - tan y - tan y + tan x 2 ( tan x - tan y )
=
=
1 + tan x tan y 1 + tan y tan x
1 + tan x tan y
1 + tan x tan y
cos (α - β )
= tan α + cot β is an identity.
cos α sin β
cos (α - β ) cos α cos β + sin α sin β cos α cos β sin α sin β
cos β sin α
=
=
+
=
+
= cot β + tan α
cos α sin β
cos α sin β
cos α sin β cos α sin β
sin β cos α
65. Verify
sin ( s + t )
= tan s + tan t is an identity.
cos s cos t
sin ( s + t ) sin s cos t + cos s sin t sin s cos t cos s sin t
sin s sin t
=
=
+
=
+
= tan s + tan t
cos s cos t
cos s cos t
cos s cos t cos s cos t cos s cos t
66. Verify
67. Verify that
sin ( x - y )
sin ( x + y )
=
tan x - tan y
is an identity
tan x + tan y
sin x cos y cos x sin y
sin x cos y cos x sin y
⋅
⋅
sin x cos y - cos x sin y
cos x cos y cos x cos y
cos x cos y cos x cos y
=
=
=
sin x cos y cos x sin y
sin x cos y cos x sin y
sin ( x + y ) sin x cos y + cos x sin y
+
⋅
+
⋅
cos x cos y cos x cos y
cos x cos y cos x cos y
sin x sin y
sin x
sin y
⋅1-1⋅
tan x - tan y
cos y
cos x cos y
cos x
=
=
=
sin x
sin y
sin x sin y
tan
x + tan y
⋅1 + 1⋅
+
cos x
cos y
cos x cos y
sin ( x - y )
68. Verify
sin ( x + y )
cos ( x - y )
=
cot x + cot y
is an identity.
1 + cot x cot y
Working with the right side, we have
cos x cos y
cos x cos y
+
+
cot x + cot y
sin x sin y
sin x sin y sin x sin y cos x sin y + sin x cos y sin ( x + y )
=
=
⋅
=
=
1 + cot x cot y 1 + cos x cos y 1 + cos x cos y sin x sin y sin x sin y + cos x cos y cos ( x - y )
sin x sin y
sin x sin y
sin ( s - t )
cos ( s - t )
sin s
is an identity.
sin t
cos t
sin t cos t
sin ( s - t ) cos ( s - t ) sin s cos t - sin t cos s cos s cos t + sin t sin s
+
=
+
sin t
cos t
sin t
cos t
69. Verify
+
=
=
=
sin s cos 2 t - sin t cos t cos s sin t cos t cos s + sin 2 t sin s sin s cos 2 t + sin s sin 2 t
+
=
sin t cos t
sin t cos t
sin t cos t
(
sin s cos 2 t + sin 2 t
sin t cos t
)=
sin s
sin t cos t
Copyright © 2013 Pearson Education, Inc.
68
Chapter 5 Trigonometric Identities
70. Verify
tan (α + β ) - tan β
1 + tan (α + β ) tan β
identity.
tan (α + β ) - tan β
1 + tan (α + β ) tan β
= tan α is an
= tan éë(α + β ) - β ùû
(b) V = 50 sin (120πt – 5.353).
= tan α
(c) 50sin (120π t - 5.353)
71. 180- 
é(sin120π t )(cos 5.353)
ù
ú
= 50 êê
ú
cos120
π
t
sin
5.353
(
)(
)
ëê
ûú
é
ù
» 50 (sin120π t )(0.5977)
ê
ú
ê
ú
cos120
π
0.8017
t
(
)(
)
ëê
ûú
» 29.89sin120π t + 40.09 cos120π t
» 30sin120π t + 40 cos120π t
72.  =  - 
73. tan  =
tan  - tan 
1 + tan  tan 
74. Substituting m1 for tan α and m2 for tan β
into the expression in exercise 69, we have
m - m1
tan β - tan α
= 2
tan θ =
1 + tan α tan β 1 + m1m2
82. z ¢ = z cos R + y sin R
75. θ » 18.4
76.  » 80.8
77. (a) 425 lb
(b) F =
=
=
=
=
81. y ¢ = y cos R - z sin R
Chapter 5 Quiz
(Sections 5.1−5.4)
0.6W sin (θ + 90)
sin12
0.6W (sin θ cos 90 + sin 90 cos θ )
sin12
0.6W (sin θ ⋅ 0 + 1⋅ cos θ )
sin12
0.6W (0 + cos θ )
sin12
0.6
W cos θ » 2.9W cos θ
sin12
(c) θ = 0
7
24
24
; tan  = - ; cot  = - ;
24
25
7
25
25
sec  = ; csc  = 24
7
1. cos  =
2.
1 + cos 2 x
sin 2 x
- 6- 2
4
4. - cos 
3.
5. (a)
-
78. (a) 408 lb
(b) θ » 46.1.
79. -20 cos
(b) -
t
16
65
63
65
(c) quadrant III
4
80. (a) The calculator should be in radian mode.
6.
-1 + tan x
1 + tan x
7. Working with the right side, we have
sin θ
sin θ csc θ + 1
=
⋅
csc θ -1 csc θ -1 csc θ + 1
sin θ csc θ + sin θ 1 + sin θ
=
=
csc2 θ -1
cot 2 θ
Copyright © 2013 Pearson Education, Inc.
Section 5.5 Double-Angle Identities
æπ
ö
æπ
ö
8. sin çç + θ ÷÷÷ - sin çç -θ ÷÷÷
çè 3
ç
ø
è3
ø
æ π
ö
π
= ççsin cos θ + cos sin θ ÷÷÷
çè 3
ø
3
æ π
ö
π
- ççsin cos θ - cos sin θ ÷÷÷
çè 3
ø
3
æ1ö
π
= 2 cos sin θ = 2 çç ÷÷÷ sin θ = sin θ
çè 2 ø
3
10. Verify
cos ( x + y ) + cos ( x - y )
sin ( x - y ) + sin ( x - y )
cos ( x + y ) + cos ( x - y )
sin ( x + y ) + sin ( x - y )
Section 5.5
=
9.
sin 2  - cos 2 
sin 4  - cos 4 
=
sin 2  - cos 2 
1
= =1
1
sin  - cos  sin  + cos 
(
2
)(
2
2
2
)
= cot x is an identity.
(cos x cos y - sin x sin y ) + (cos x cos y + sin x sin y ) 2 cos x cos y cos x
=
=
= cot x
(sin x cos y + cos x sin y ) + (sin x cos y - cos x sin y ) 2sin x cos y sin x
Double-Angle Identities
1. C
16. cos  = -
30
6
; sin  =
6
6
17.
2. E
3. B
(sin x + cos x)2 = sin 2 x + 2sin x cos x + cos 2 x
4. A
(
)
= sin 2 x + cos 2 x + 2sin x cos x
5. F
= 1 + sin 2 x
6. D
4 21
17
; sin 2 = 25
25
120
119
8. sin 2 = ;cos 2 =
169
169
7. cos 2 =
16 4
3
9. cos 2 x = - ; sin 2 x =
= .
25 5
5
8
15
10. cos 2 x = - ; sin 2 x =
17
17
39
4 55
11. cos 2 = ;sin 2 = 49
49
12. cos 2 = 13. cos  =
69
19
2 66
;sin 2 =
25
25
2 5
5
; sin  =
5
5
14. cos  = -
18. Work with the right side.
sec2 x + sec4 x
2 + sec 2 x - sec4 x
1
1
+
2
4
= cos x cos x
1
1
2+
2
cos x cos 4 x
1
=
=
=
2
14
; sin  = 4
4
42
102
; sin  =
15. cos  = 12
12
=
1
+
4
cos x cos 4 x ⋅ cos x
1
1
cos 4 x
2+
cos 2 x cos 4 x
2
cos 2 x + 1
2 cos 4 x + cos 2 x -1
cos 2 x + 1
(2 cos2 x -1)(cos2 x +1)
1
2
2 cos x -1
Copyright © 2013 Pearson Education, Inc.
=
1
= sec 2 x
cos 2 x
70
Chapter 5 Trigonometric Identities
(
2
19. (cos 2 x + sin 2 x)
= cos 2 2 x + 2 cos 2 x sin 2 x + sin 2 2 x
(
)
= cos 2 2 x + sin 2 2 x + 2 cos 2 x sin 2 x
= 1 + sin 4 x
2
20. (cos 2 x + sin 2 x)
= cos 2 2 x - 2 cos 2 x sin 2 x + sin 2 2 x
(
)
= cos 2 2 x + sin 2 2 x - 2 cos 2 x sin 2 x
27. Work with the right side.
cos θ sin θ
cot θ - tan θ =
sin θ cos θ
cos θ cos θ sin θ sin θ
=
⋅
⋅
sin θ cos θ cos θ sin θ
= 1- sin 4 x
=
21. tan 8θ - tan 8θ tan 2 4θ
(
2
= tan 8θ 1- tan 4θ
=
)
=
2 tan 4θ
1- tan
2
1- tan 2 4θ ) = 2 tan 4θ
(
4θ
22. Working with the right side, we have
sin x
sin x
2⋅
2⋅
2
2 tan x
cos x =
cos x ⋅ cos x
=
1 + tan 2 x
sin 2 x
sin 2 x cos 2 x
1+
1
+
cos 2 x
cos 2 x
2sin x cos x
=
= 2sin x cos x
cos 2 x + sin 2 x
28. cot 4θ =
=
=
2 cos 2 θ -1
= cos 2θ
1
=
2 tan θ
1- tan 2 θ
(
2 cos 2 θ - sin 2 θ
2sin θ cos θ
) = 2 cos 2θ
sin 2θ
1- tan 2 2θ
2 tan 2θ
29. tan x + cot x =
sin x sin x cos x cos x
⋅
+
⋅
cos x sin x sin x cos x
sin 2 x + cos 2 x
1
=
cos x sin x
cos x sin x
2
2
=
=
= 2 csc 2 x
2 cos x sin x sin 2 x
=
30. Work with the right side.
1- tan 2 x
1 + tan 2 x
24. Working with the right side, we have
-2 tan θ
2 tan θ
2 tan θ
==2
2
sec θ - 2
tan 2 θ -1
1 + tan θ - 2
(
cos 2 θ - sin 2 θ
sin θ cos θ
1
1
1
=
=
2 tan 2θ
tan 4θ tan 2 (2θ )
1- tan 2 2θ
= sin 2 x
23. Working with the right side, we have
1
1
222
2
2
2 - sec θ
cos θ =
cos 2 θ ⋅ cos θ
=
1
1
sec 2 θ
cos 2 θ
cos 2 θ
cos 2 θ
)
2
1 + cos 2 x 1 + 2 cos x -1
2 cos 2 x
=
=
26.
sin 2 x
2sin x cos x
2sin x cos x
cos x
=
= cot x
sin x
)
= tan 2θ
25. sin 4 x = sin 2 (2 x) = 2sin 2 x cos 2 x
= 2 (2sin x cos x) cos 2 x
= 4sin x cos x cos 2 x
Copyright © 2013 Pearson Education, Inc.
1=
1+
=
sin 2 x
cos 2 x =
sin 2 x
cos 2 x
11+
cos 2 x - sin 2 x
cos 2 x + sin 2 x
sin 2 x
2
cos 2 x ⋅ cos x
sin 2 x cos 2 x
cos 2 x
=
cos 2 x - sin 2 x
1
= cos 2 x - sin 2 x = cos 2 x
Section 5.5 Double-Angle Identities
31.
71
32.
æ 2 tan x ÷ö
÷
1 + tan x tan 2 x = 1 + tan x çç
çè1- tan 2 x ø÷÷
= 1+
cos A sin A
cot A - tan A
= sin A cos A
cot A + tan A cos A + sin A
sin A cos A
cos A sin A
sin A cos A
= sin A cos A ⋅
cos A sin A sin A cos A
+
sin A cos A
2 tan 2 x
1- tan 2 x
1- tan 2 x) + 2 tan 2 x
(
=
1- tan 2 x
=
2
1 + tan x
2
1- tan x
1+
=
1=
1+
=
1-
sin 2 x
2
=
2
=
cos x
sin 2 x
cos x
2
sin x
cos 2 A - sin 2 A
cos 2 A + sin 2 A
cos 2 A - sin 2 A
1
= cos 2 A - sin 2 A = cos 2 A
2
cos x ⋅ cos x
sin 2 x cos 2 x
2
cos 2 x
cos 2 x + sin 2 x
2
2
cos x - sin x
= sec 2 x
=
1
cos 2 x
33. Verify sin 2 A cos 2 A = sin 2 A - 4 sin 3 A cos A is an identity.
(
)
sin 2 A cos 2 A = (2sin A cos A) 1- 2sin 2 A = 2sin A cos A - 4sin 3 A cos A = sin 2 A - 4sin 3 A cos A
34. Verify sin 4 x = 4 sin x cos x - 8sin 3 x cos x is an identity.
(
)
sin 4 x = sin 2 (2 x) = 2sin 2 x cos 2 x = 2 (2sin x cos x) 1- 2sin 2 x = 4sin x cos x - 8sin 3 x cos x
35. Verify tan (θ - 45) + tan (θ + 45) = 2 tan 2θ is an identity.
tan (θ - 45) + tan (θ + 45) =
=
=
tan θ - tan 45
tan θ + tan 45
tan θ -1 tan θ + 1 tan θ -1 tan θ + 1
+
=
+
=
1 + tan θ tan 45 1- tan θ tan 45 1 + tan θ 1- tan θ
tan θ + 1 tan θ -1
2
2
( tan θ -1)2 - ( tan θ + 1)2 ( tan θ - 2 tan θ + 1) - ( tan θ + 2 tan θ + 1)
=
( tan θ + 1)( tan θ -1)
tan 2 θ -1
2 (2 tan θ )
-4 tan θ
4 tan θ
tan 2 θ -1
=
1- tan 2 θ
=
1- tan 2 θ
= 2 tan 2θ
æπ
ö
36. Verify cot θ tan (θ + π ) - sin (π -θ ) cos çç -θ ÷÷÷ = cos2 θ is an identity.
çè 2
ø
æπ
ö
cot θ tan (θ + π ) - sin (π -θ ) cos çç -θ ÷÷÷
çè 2
ø
æ π
ö
π
tan θ + tan π
- (sin π cos θ - cos π sin θ )ççcos cos θ + sin sin θ ÷÷÷
ç
è
ø
1- tan θ tan π
2
2
tan θ + 0
tan θ
= cot θ ⋅
- (0 ⋅ cos θ - (-1) sin θ )(0 ⋅ cos θ + 1⋅ sin θ ) = cot θ ⋅
- (0 + sin θ )(0 + sin θ )
1- tan θ (0)
1- 0
= cot θ ⋅
= cot θ tan θ - sin 2 θ = 1- sin 2 θ = cos 2 θ
Copyright © 2013 Pearson Education, Inc.
72
Chapter 5 Trigonometric Identities
37.
3
2
59. sin
38.
3
3
60.
39.
3
2
40.
2
2
41. 42.
43.

2
- sin

6
5
5
cos 5 x + cos x
2
2
61. 3cos x - 3cos 9 x
62. 4 cos 2 x - 4 cos16 x
63. -2 sin 3 x sin x
64. 2 cos 6.5 x cos1.5 x
65. -2 sin11.5 cos 36.5
2
2
66. 2 cos 98.5 sin 3.5
67. 2 cos 6 x cos 2 x
2
4
68. 2 cos 6 x sin 3 x
1
tan102
2
69. a = –885.6; c = 885.6;  = 240
70. (a) Graph
W = VI = éë163sin (120π t )ùû éë1.23sin (120π t )ùû
over the interval 0 ≤ t ≤ 0.05.
1
44.
tan 68
4
45.
1
cos 94.2
4
46.
1
sin 59
16
47. - cos
4
5
48. cos 4x
49. 4 sin x cos3 x - 4 sin 3 x cos x
(b) The minimum wattage is 0 and the
maximum wattage is 200.49 watts
50. cos 3 x = -3cos x + 4 cos3 x
51. tan 3 x =
(c) a = –100.245; ω = 240π ; c = 100.245
3 tan x - tan 3 x
2
1 - 3 tan x
(d) The graphs of
W = éë163sin (120π t )ùû éë1.23sin (120π t )ùû
52. cos 4 x = 8 cos 4 x - 8 cos 2 x + 1
53. equivalent; cos 4 x - sin 4 x = cos 2 x
54. equivalent;
55. equivalent;
4 tan x cos 2 x - 2 tan x
1- tan 2 x
2 tan x
2 - sec2 x
= sin 2 x
= tan 2 x
cot 2 x -1
= cot 2 x
2 cot x
57. sin160 - sin 44
56. equivalent;
and W = -100.245cos 240π t +100.245
are the same.
(e) 100 watts
58. sin 225 + sin 55
Copyright © 2013 Pearson Education, Inc.
Section 5.6 Half-Angle Identities
Section 5.6
Half-Angle Identities
18. Show
3 - 2 2 = 2 -1.
1. the negative square root.
3- 2 2 = 3- 2 2
2. the positive square root.
3 - 2 2 = 2 - 2 2 +1
3. the positive square root.
3- 2 2 =
4. the negative square root.
2
- 2 2 + 12
5. C
(
6. A
If a 2 = b2 , then a = b. Thus,
3- 2 2
7. D
9. F
19.
10
4
20.
13
4
10. B
2+ 2
2
12. -
21. 3
2- 3
2
22. -
13. 2 - 3
14.
- 2+ 3
2
15.
- 2+ 3
2
16.
=
(
17. To find sin 7.5º, you could use the half-angle
formulas for sine and cosine as follows:
1- cos15
and
sin 7.5 =
2
1+
1 + cos 30
=
2
2
3
2
2+ 3
2+ 3
=
=
4
2
=
50 -10 5
10
24.
50 -15 10
10
1- cos15
=
2
1-
26.
5
5
27.
5
5
28. -
3
2
29. -
42
12
30. -
6
6
31. 0.127
Thus,
sin 7.5 =
23.
25. - 7
2- 3
2
cos15 =
5
5
2+ 3
2
2
2- 2 + 3
2- 2 + 3
=
4
2
32. 2.646
33. sin 20
34. cos 38
35. tan 73.5
36. cot 82.5
37. tan 29.87
Copyright © 2013 Pearson Education, Inc.
2
)
2 -1
3 - 2 2 = 2 -1.
8. E
11.
)
2
( 2)
73
74
Chapter 5 Trigonometric Identities
Work with the right side.
x
x 1 + cos x 1- cos x
cos 2 - sin 2 =
2
2
2
2
2 cos x
=
= cos x
2
sin 2 x
x
x
Since
= cos x = cos 2 - sin 2 , the
2sin x
2
2
statement has been verified.
38. tan 79.1
39. cos 9 x
40. cos10
41. tan 4
42. tan
5A
2
43. cos
x
8
44. sin
3
10
45. sec 2
æ 1- cos x ö÷2
2
2
2 x
49.
- tan
=
- çç
÷÷
1 + cos x
2 1 + cos x çèç 1 + cos x ÷ø
2
1- cos x
1 + cos x 1 + cos x
2 -1 + cos x
=
1 + cos x
1 + cos x
=
=1
1 + cos x
=
x
1
1
=
=
2 cos 2 x æ 1 + cos x ö2
÷÷
ç
2 ç
÷
çç
2
è
ø÷
=
1
2
=
1 + cos x 1 + cos x
2
50. tan
2
æ
÷÷ö
çç
(1 + cos x)2
x ç 1 ÷÷
1
46. cot 2 = çç
÷÷ =
=
2 çç tan x ÷÷
sin 2 x
æ sin x ö÷2
÷ø
çç
÷
ççè
÷
2
çè1 + cos x ÷ø
47. Work with the left side.
2
x æ 1- cos x ö÷÷
1- cos x
sin 2 = ççç
÷÷ =
2 çè
2
2
ø
Work with the right side.
sin x
- sin x
tan x - sin x cos x
=
sin x
2 tan x
2⋅
cos x
sin x
- sin x
cos x
= cos x
⋅
sin x
cos x
2⋅
cos x
sin x - cos x sin x
=
2sin x
sin x (1- cos x ) 1- cos x
=
=
2sin x
2
x
1
cos
x
tan
x
sin
x
Since sin 2 =
=
, the
2
2
2 tan x
statement has been verified.
θ
2
sin θ
sin θ 1- cos θ
=
⋅
1 + cos θ 1 + cos θ 1- cos θ
sin θ (1- cos θ ) sin θ (1- cos θ )
=
=
1- cos 2 θ
sin 2 θ
1- cos θ
1
cos θ
=
=
sin θ
sin θ sin θ
= csc θ - cot θ
=
51.
1- tan 2
æ sin θ ö÷2
sin 2 θ
= 1- çç
=
1
÷
çè1 + cos θ ÷ø
2
(1 + cos θ )2
θ
48. Work with the left side.
sin 2 x 2sin x cos x
=
= cos x
2sin x
2sin x
Copyright © 2013 Pearson Education, Inc.
=
=
=
2
(1 + cos θ ) - sin 2 θ
2
(1 + cos θ )
1 + 2 cos θ + cos 2 θ - sin 2 θ
2
(1 + cos θ )
(
1 + 2 cos θ + cos 2 θ - 1- cos 2 θ
2
(1 + cos θ )
=
1 + 2 cos θ + 2 cos 2 θ -1
=
2 cos 2 θ + 2 cos θ
=
(1 + cos θ )2
(1 + cos θ )2
2 cos θ (1 + cos θ )
2
(1 + cos θ )
=
2 cos θ
1 + cos θ
)
Section 5.6 Half-Angle Identities
52. Working with the right side, we have
1- cos x
1- tan 2 2x 1- 1 + cos x
=
1 + tan 2 2x 1 + 1- cos x
1 + cos x
1- cos x
11 + cos x ⋅ 1 + cos x
=
1- cos x 1 + cos x
1+
1 + cos x
+
1
( cos x) - (1- cos x)
=
(1 + cos x) + (1- cos x)
=
53. tan
54. tan
2
æ xö
sin x
= tan çç ÷÷÷
ç
è 2ø
1 + cos x
tan
cot
x + cot x
2
2
x - tan x
2
2
58. equivalent; 1- 8sin 2
59. 106
60. 84
61. m = 2
69. AD 2 = AC 2 + CD 2 
2
)
(
= 8+ 4 3 =
= 1+ 4 + 4 3 + 3
6+ 2
2
)
70.
6+ 2
4
6- 2
4
72. 2 - 3
71.
73. cos18 =
10 + 2 5
4
74. Use the result from exercise 73.
(-5 + 3 5)(
10 - 2 5
x
x
cos 2 = cos 2 x
2
2
tan18 =
)
20
Alternative solution
= sec x

AD = 6 + 2
(5 - 5)(
10 - 2 5
)
20
Use a calculator to show that the two forms
for tan 18° are equal.
75. Use the result from exercise 73.
cot18 =
62. m » 3.9
( 10 + 2 5 )(
)
5 +1
4
76. Use the result from exercise 73.
θ
R -b
63. (a) cos =
2
R

b
(b) tan =
4 50
64. 54
68. 2 + 3
tan18 =
1- cos x
x
56. equivalent;
= tan
sin x
2
57. equivalent;
67. The sum of the measures of angles DAB and
ADB is 180º − 150º = 30º.
mDAB = mADB , so the measure of each
is 15º.
(
1
1 + cos A
=
sin A
sin A
1 + cos A
55. equivalent;
66. mABD = 150 because it is the supplement
of a 30º angle.
AD 2 = 12 + 2 + 3
A
sin A
1
=
=

2 1 + cos A cot A
A
cot =
2
65. AB = BD because they are both radii of the
circle.
2 cos x
= cos x
2
A
sin A
sin A 1- cos A
=
=
⋅
2 1 + cos A 1 + cos A 1- cos A
sin A(1- cos A) sin A(1- cos A)
=
=
1- cos 2 A
sin 2 A
1- cos A
=
sin A
75
sec18 =
(5 - 5)
10 - 2 5
10
Alternate solution
50 + 10 5
5
Use a calculator to show that the two forms
for sec 18° are equal.
sec18 =
Copyright © 2013 Pearson Education, Inc.
76
Chapter 5 Trigonometric Identities
Summary Exercises on Verifying
Trigonometric Identities
77. csc18 = 5 + 1
78. cos 72 =
5 -1
4
79. sin 72 =
10 + 2 5
4
80. tan 72 =
81. tan 72 =
1. Verify tan θ + cot θ = sec θ csc θ is an
identity.
sin θ cos θ
tan θ + cot θ =
+
cos θ sin θ
( 10 + 2 5 )(
)
=
5 +1
10 - 2 5
20
( 10 + 2 5 )(
tan 72 =
(5 - 5 )
)
5 +1
10
82.
csc 72 =
10 - 2 5
=
) or
4
csc 72 =
sin 2 θ + cos 2 θ
1
=
cos θ sin θ
cos θ sin θ
1
1
=
⋅
= sec θ csc θ
cos θ sin θ
4
(-5 + 3 5 )(
or
2. Verify csc θ cos 2 θ + sin θ = csc θ is an
identity.
1
⋅ cos 2 θ + sin θ
csc θ cos 2 θ + sin θ =
sin θ
=
50 -10 5
5
cos 2 θ sin 2 θ
+
sin θ
sin θ
cos 2 θ + sin 2 θ
1
=
sin θ
sin θ
= csc θ
=
83. sec 72 = 1 + 5
84. sin162 =
sin 2 θ
cos 2 θ
+
cos θ sin θ cos θ sin θ
x
= csc x - cot x is an identity.
2
Starting on the right side, we have
1
cos x 1 - cos x
x
csc x - cot x =
=
= tan
sin x sin x
sin x
2
5 -1
4
3. Verify tan
4. Verify sec (π - x) = - sec x is an identity.
sec (π - x ) =
1
1
1
1
1
=
=
=
== - sec x
cos (π - x ) cos π cos x + sin π sin x (-1) cos x + (0) sin x - cos x + 0
cos x
sin t
1- cos t
is an identity.
=
1 + cos t
sin t
sin t
sin t 1- cos t sin t (1- cos t ) sin t (1- cos t ) 1- cos t
=
⋅
=
=
=
1 + cos t 1 + cos t 1- cos t
sin t
1- cos 2 t
sin 2 t
5. Verify
1- sin t
1
is an identity.
=
cos t
sec t + tan t
1
1
1
cos t
cos t 1- sin t
=
=
=
=
⋅
1
sin t
1 + sin t 1 + sin t 1 + sin t 1- sin t
sec t + tan t
+
cos t cos t
cos t
cos t (1- sin t ) cos t (1- sin t ) 1- sin t
=
=
=
cos t
1- sin 2 t
cos 2 t
6. Verify
Copyright © 2013 Pearson Education, Inc.
Summary Exercises on Verifying Trigonometric Identities
2 tan θ
7. Verify sin 2θ =
is an identity.
1 + tan 2 θ
Starting on the right side, we have
sin θ
2⋅
2 tan θ
2 tan θ
sin θ cos 2 θ
=
= cos θ = 2 ⋅
⋅
= 2sin θ cos θ = sin 2θ
1
cos θ
1
1 + tan 2 θ sec2 θ
cos 2 θ
8. Verify
2
x
- tan 2 = 1 is an identity.
1 + cos x
2
2 (1 + cos x )
æ sin x ö÷2
2
2
2
sin 2 x
sin 2 x
x
- tan 2 =
- çç
=
=
÷
1 + cos x
2 1 + cos x çè1 + cos x ø÷
1 + cos x (1 + cos x )2 (1 + cos x )2 (1 + cos x )2
=
=
2 + 2 cos x - sin 2 x
2
(1 + cos x)
cos 2 x + 2 cos x + 1
(1 + cos x)2
9. Verify cot θ - tan θ =
cot θ - tan θ =
=
=
=
(
2 + 2 cos x - 1- cos 2 x
2
2
(1 + cos x)
(1 + cos x)2
(1 + cos x)2
) = 2 + 2 cos x -1 + cos2 x
(1 + cos x)
=1
2 cos 2 θ -1
is an identity.
sin θ cos θ
(
2
2
cos θ sin θ
cos 2 θ
sin 2 θ
cos 2 θ - sin 2 θ cos θ - 1- cos θ
=
=
=
sin θ cos θ sin θ cos θ sin θ cos θ
sin θ cos θ
sin θ cos θ
)
cos 2 θ -1 + cos 2 θ 2 cos 2 θ -1
=
sin θ cos θ
sin θ cos θ
1
1
+
= 2 cot t csc t is an identity.
sec t -1 sec t + 1
1
1
1
1
1
cos t
1
cos t
cos t
cos t
+
=
+
=
⋅
+
⋅
=
+
1
1
1
1
sec t -1 sec t + 1
-1
+1
-1 cos t
+ 1 cos t 1- cos t 1 + cos t
cos t
cos t
cos t
cos t
10. Verify
=
=
11. Verify
sin ( x + y )
cos ( x - y )
=
cos t 1 + cos t
cos t 1- cos t cos t + cos 2 t cos t - cos 2 t
⋅
+
⋅
=
+
1- cos t 1 + cos t 1 + cos t 1- cos t
1- cos 2 t
1- cos2 t
cos t + cos 2 t + cos t - cos2 t
2
1- cos t
=
2 cos t
2
1- cos t
=
2 cos t
2
sin t
= 2⋅
cos t 1
⋅
= 2 cot t csc t
sin t sin t
cot x + cot y
is an identity.
1 + cot x cot y
1
sin x cos y cos x sin y
+
sin x cos y + cos x sin y sin x cos y + cos x sin y cos x cos y
cos x cos y cos x cos y
=
=
⋅
=
1
cos x cos y sin x sin y
cos ( x - y ) cos x cos y + sin x sin y cos x cos y + sin x sin y
+
cos x cos y
cos x cos y cos x cos y
sin x sin y
+
cot x + cot y
cos x cos y
=
=
sin x sin y 1 + cot x cot y
1+
⋅
cos x cos y
sin ( x + y )
Copyright © 2013 Pearson Education, Inc.
77
78
Chapter 5 Trigonometric Identities
12. Verify 1- tan 2
θ
=
2
2 cos θ
is an identity.
1 + cos θ
2
æ sin θ ö÷2
(1 + cos θ )
sin 2 θ
sin 2 θ
1 + 2 cos θ + cos 2 θ
sin 2 θ
1- tan
= 1- çç
=
1
=
=
÷
2
2
2
2
2
çè1 + cos θ ÷ø
2
(1 + cos θ ) (1 + cos θ ) (1 + cos θ )
(1 + cos θ )
(1 + cos θ )
2θ
=
1 + 2 cos θ + cos 2 θ - sin 2 θ
=
2 cos θ + 2 cos 2 θ
2
(1 + cos θ )
2
(1 + cos θ )
=
=
(
2 cos θ + cos 2 θ + 1- sin 2 θ
2
(1 + cos θ )
2 cos θ (1 + cos θ )
2
(1 + cos θ )
=
) = 2 cosθ + cos2 θ + cos2 θ
2
(1 + cos θ )
2 cos θ
1 + cos θ
sin θ + tan θ
= tan θ is an identity.
1 + cos θ
sin θ
sin θ
sin θ +
sin θ +
sin θ + tan θ
cos
cos
θ
θ ⋅ cos θ = sin θ cos θ + sin θ = sin θ (cos θ + 1) = sin θ = tan θ
=
=
1 + cos θ
1 + cos θ
1 + cos θ
cos θ
cos θ (1 + cos θ )
cos θ (1 + cos θ ) cos θ
13. Verify
14. Verify csc4 x - cot 4 x =
4
4
csc x - cot x =
=
1
sin 4 x
1 + cos 2 x
1- cos 2 x
-
cos 4 x
sin 4 x
1 + cos 2 x
sin 2 x
=
=
is an identity.
1- cos 4 x
sin 4 x
1 + cos 2 x)(1- cos 2 x) (1 + cos 2 x)(sin 2 x)
(
=
=
sin 4 x
sin 4 x
1 + cos 2 x
1- cos 2 x
x
2 is an identity.
15. Verify cos x =
2 x
1 + tan
2
1- tan 2
2
2
2
(1- cos x)
(1- cos x)
x 1- æç1- cos x ö÷
112
÷÷
2
çèç
2
2
ø
sin x
sin x =
sin 2 x ⋅ sin x = sin x - (1- cos x )
2=
=
x
æ1- cos x ö÷2
(1- cos x)2
(1- cos x)2 sin 2 x sin 2 x + (1- cos x)2
1 + tan 2
çç
1
+
÷
1+
1+
2
çè sin x ÷ø
sin 2 x
sin 2 x
1- tan 2
2
2
(
)
sin 2 x -1 + 2 cos x - cos 2 x (1- cos x) -1 + 2 cos x - cos x
=
=
=
2
2
sin 2 x + (1- 2 cos x + cos 2 x) sin x + 1- 2 cos x + cos x
(sin 2 x + cos2 x) +1- 2 cos x
sin 2 x - 1- 2 cos x + cos 2 x
=
1- cos 2 x -1 + 2 cos x - cos 2 x 2 cos x - 2 cos 2 x 2 cos x (1- cos x )
=
=
= cos x
1 + 1- 2 cos x
2 - 2 cos x
2 (1- cos x )
16. Verify cos 2 x =
2 - sec 2 x
sec 2 x
2 - sec 2 x
2=
sec 2 x
1
is an identity. Starting on the right side, we have
2
2
cos 2 x ⋅ cos x = 2 cos x -1 = 2 cos 2 x -1 = cos 2 x
1
1
cos 2 x
cos 2 x
Copyright © 2013 Pearson Education, Inc.
Summary Exercises on Verifying Trigonometric Identities
17. Verify
tan 2 t + 1
tan t csc2 t
79
= tan t is an identity.
sin 2 t
sin 2 t
sin 2 t
+
+
+1
1
1
2
2
2
tan 2 t + 1
cos 2 t sin t sin 3 t + cos 2 t sin t
= cos t
= cos t
= cos t
⋅
=
2
sin t
1
1
1
cos t
t
t
tan t csc2 t
cos
sin
⋅ 2
cos t sin t
cos t sin t
cos t sin t
=
18. Verify
(
sin t sin 2 t + cos 2 t
cos t
) = sin t (1) = sin t = tan t
cos t
cos t
sin s
1 + cos s
+
= 2 csc s is an identity.
1 + cos s
sin s
(
2
sin 2 s + 1 + 2 cos s + cos 2 s
sin 2 s + (1 + cos s )
(1 + cos s )2
sin s
1 + cos s
sin 2 s
+
=
+
=
=
1 + cos s
sin s
sin s (1 + cos s ) sin s (1 + cos s )
sin s (1 + cos s )
sin s (1 + cos s )
=
19. Verify tan 4θ =
(1- cos2 s) +1 + 2 cos s + cos2 s =
sin s (1 + cos s )
2 (1 + cos s )
2 + 2 cos s
2
=
=
= 2 csc s
sin s (1 + cos s ) sin s (1 + cos s ) sin s
2 tan 2θ
is an identity.
2 - sec2 2θ
sin 2θ
sin 2θ
2⋅
2⋅
2
é
ù
2 tan 2θ
cos
2
θ
cos
2θ ⋅ cos 2θ = 2sin 2θ cos 2θ = sin ë 2 (2θ )û = sin 4θ = tan 4θ
=
=
1
cos éë 2 (2θ )ùû cos 4θ
2 - sec2 2θ 2 - 1
cos2 2θ
2 cos 2 2θ -1
22
2
cos 2θ
cos 2θ
æx π ö
20. Verify tan çç + ÷÷÷ = sec x + tan x is an identity.
çè 2 4 ø
sin x
sin x
x
x
x
π
+1
+1
tan + 1
tan + 1
æ x π ö÷ tan 2 + tan 4
1 + cos x
2
2
tan çç + ÷÷ =
=
=
= 1 + cos x
= 1 + cos x
⋅
çè 2 4 ø
æ
ö
x
x
x
x
sin
sin
π
x
1 + cos x
1- tan tan
111- çç tan ÷÷÷(1) 1- tan
çè
2
4
2
1 + cos x
1 + cos x
2ø
sin x + (1 + cos x )
=
=
sin x + 1 + cos x sin x + 1 + cos x cos x sin x cos x + cos x + cos 2 x
⋅
=
=
1 + cos x - sin x 1 + cos x - sin x cos x
cos x (1 + cos x - sin x )
(1 + cos x) - sin x
cos x (1 + sin x ) + (1 + sin x )(1- sin x ) (1 + sin x )(cos x + 1- sin x )
=
=
cos x (1 + cos x - sin x )
cos x (1 + cos x - sin x )
=
1 + sin x
1
sin x
=
+
= sec x + tan x
cos x
cos x cos x
cot s - tan s cos s - sin s
is an identity.
=
cos s + sin s
sin s cos s
cos s sin s
cos s sin s
cot s - tan s
sin s cos s
cos 2 s - sin 2 s
= sin s cos s = sin s cos s ⋅
=
cos s + sin s
cos s + sin s
cos s + sin s sin s cos s (cos s + sin s ) sin s cos s
21. Verify
=
(cos s + sin s )(cos s - sin s ) cos s - sin s
=
sin s cos s
(cos s + sin s ) sin s cos s
Copyright © 2013 Pearson Education, Inc.
)
80
Chapter 5 Trigonometric Identities
tan θ - cot θ
= 1- 2 cos 2 θ is an identity.
tan θ + cot θ
sin θ cos θ
sin θ cos θ
tan θ - cot θ
cos θ sin θ sin 2 θ - cos 2 θ sin 2 θ - cos 2 θ
= cos θ sin θ = cos θ sin θ ⋅
=
=
sin θ cos θ
sin θ cos θ cos θ sin θ sin 2 θ + cos 2 θ
tan θ + cot θ
1
+
+
cos θ sin θ
cos θ sin θ
22. Verify
(
)
= sin 2 θ - cos 2 θ = 1- cos 2 θ - cos 2 θ = 1- 2 cos 2 θ
23. Verify
tan ( x + y ) - tan y
1 + tan ( x + y ) tan y
= tan x is an identity.
tan x + tan y
tan x + tan y
- tan y
- tan y
1- tan x tan y
1- tan x tan y
1- tan x tan y
=
=
⋅
1 + tan ( x + y ) tan y 1 + tan x + tan y ⋅ tan y 1 + tan x + tan y ⋅ tan y 1- tan x tan y
1- tan x tan y
1- tan x tan y
tan ( x + y ) - tan y
=
tan x + tan y - tan y (1- tan x tan y )
1- tan x tan y + ( tan x + tan y ) tan y
(
tan x 1 + tan 2 y
=
24. Verify 2 cos 2
1 + tan 2 y
=
tan x + tan x tan 2 y
1- tan x tan y + tan x tan y + tan 2 y
) = tan x
x
tan x = tan x + sin x is an identity.
2
æ 1 + cos x ö÷2 sin x
x
1 + cos x sin x
sin x
sin x
÷÷ ⋅
2 cos
tan x = 2 ççç
= 2⋅
⋅
= (1 + cos x )⋅
=
+ sin x = tan x + sin x
÷
çè
2
2
2
cos x
cos x cos x
ø cos x
2
25. Verify
cos 4 x - sin 4 x
2
cos x
cos 4 x - sin 4 x
cos 2 x
=
=
= 1- tan 2 x is an identity.
(cos2 x + sin 2 x)(cos2 x - sin 2 x) = (1)(cos2 x - sin 2 x) = cos2 x - sin 2 x
cos 2 x
cos 2 x
2
cos x
-
sin 2 x
2
cos x
cos 2 x
cos 2 x
= 1- tan 2 x
csc t + 1
2
= (sec t + tan t ) is an identity.
csc t -1
1
1
+1
+1
csc t + 1 sin t
sin t 1 + sin t 1 + sin t 1 + sin t
=
= sin t
⋅
=
=
⋅
1
1
csc t -1
-1
-1 sin t 1- sin t 1- sin t 1 + sin t
sin t
sin t
26. Verify
=
(1 + sin t )2
1- sin 2 t
=
(1 + sin t )2
cos 2 t
2
æ1 + sin t ÷ö2 æ 1
sin t ÷ö
2
çç
= çç
=
+
= (sec t + tan t )
÷
÷
çè cos t ÷ø
çè cos t cos t ÷ø
Copyright © 2013 Pearson Education, Inc.
Summary Exercises on Verifying Trigonometric Identities
27. Verify
(
2 sin x - sin 3 x
cos x
(
3
2 sin x - sin x
cos x
81
) = sin 2 x is an identity.
) = 2sin x (1- sin 2 x) = 2sin x cos2 x = 2sin x cos x = sin 2 x
cos x
cos x
1
x 1
x
cot - tan = cot x is an identity.
2
2 2
2
1
x 1
x 1 1
1
x 1
1
1 1- cos x 1 + cos x 1- cos x
cot - tan = ⋅
- tan = ⋅
- ⋅
=
2
2 2
2 2 tan x 2
2 2 sin x
2 sin x
2sin x
2sin x
2
1 + cos x
1 + cos x - (1- cos x ) 1 + cos x -1 + cos x 2 cos x cos x
=
=
=
=
= cot x
2sin x
2sin x
2sin x sin x
28. Verify
29. Verify sin (60 + x ) + sin (60- x ) = 3 cos x is an identity.
sin (60 + x ) + sin (60- x ) = (sin 60 cos x + cos 60 sin x ) + (sin 60 cos x - cos 60 sin x )
æ 3 ö÷
= 2sin 60 cos x = 2 ççç ÷÷ cos x = 3 cos x
çè 2 ÷ø
30. Verify sin (60- x) - sin (60 + x) = - sin x is an identity.
sin (60- x) - sin (60 + x) = (sin 60 cos x - cos 60 sin x) - (sin 60 cos x + cos 60 sin x)
æ1ö
= -2 cos 60 sin x = -2 çç ÷÷÷ sin x = - sin x
çè 2 ø
31. Verify
cos ( x + y ) + cos ( y - x)
sin ( x + y ) - sin ( y - x )
cos ( x + y ) + cos ( y - x)
sin ( x + y ) - sin ( y - x)
=
= cot x is an identity.
(cos x cos y - sin x sin y ) + (cos y cos x + sin y sin x) 2 cos x cos y cos x
=
=
= cot x
(sin x cos y + cos x sin y ) -(sin y cos x - cos y sin x) 2 cos y sin x sin x
32. Verify sin x + sin 3 x + sin 5 x + sin 7 x = 4 cos x cos 2 x sin 4 x is an identity.
sin x + sin 3x + sin 5 x + sin 7 x = (sin x + sin 3x) + (sin 5 x + sin 7 x)
æ x + 3x ö÷ æ x - 3x ö÷
æ 5 x + 7 x ÷ö æ 5 x - 7 x ÷ö
= 2sin çç
cos çç
+ 2sin çç
cos ç
÷
÷
÷
÷
çè 2 ÷÷ø ççè 2 ÷÷ø
èç 2 ø èç 2 ø
= 2sin 2 x cos (-x ) + 2sin 6 x cos (-x )
æ 2 x + 6 x ÷ö æ 2 x - 6 x ö÷
= 2 cos (-x ) (sin 2 x + sin 6 x) = 2 cos x ⋅ 2sin çç
cos ç
çè 2 ÷÷ø ççè 2 ÷÷ø
= 4 cos x sin 4 x cos (-2 x ) = 2 cos x cos 2 x sin 4 x
33. Verify sin 3 θ + cos3 θ + sin θ cos 2 θ + sin 2 θ cos θ = sin θ + cos θ is an identity.
(
) (
)
= sin θ (sin 2 θ + cos 2 θ ) + cos θ (cos 2 θ + sin 2 θ )
= (sin θ + cos θ )(sin 2 θ + cos 2 θ ) = sin θ + cos θ
sin 3 θ + cos3 θ + sin θ cos 2 θ + sin 2 θ cos θ = sin 3 θ + sin θ cos 2 θ + cos3 θ + sin 2 θ cos θ
Copyright © 2013 Pearson Education, Inc.
82
Chapter 5 Trigonometric Identities
34. Verify
cos x + sin x cos x - sin x
= 2 tan 2 x
cos x - sin x cos x + sin x
2
2
cos x + sin x cos x - sin x (cos x + sin x ) - (cos x - sin x )
=
cos x - sin x cos x + sin x
(cos x + sin x)(cos x - sin x )
cos 2 x + 2sin x cos x + sin 2 x) - (cos 2 x - 2sin x cos x + sin 2 x )
(
=
cos 2 x - sin 2 x
4sin x cos x
2sin 2 x
=
=
= 2 tan 2 x
2
2
cos 2 x
cos x - sin x
Chapter 5
Review Exercises
1. B
16. (a) sin
4. F
5. D
6. E
(b) sin
7. 1
10.
cos 2 
sin 
12.
tan
cos2 
17. I
1 + cos 
sin 
18. B
cos 
sin 
20. A
1
22. C

12

12
=
2- 3
;
2
=
2+ 3
;
2
= 2- 3
19. H
21. G
2
sin  cos 2 
23. J
4
4
3
13. sin x = - ; tan x = - ;cot x = 5
3
4
24. D
41
4
4 41
; cot x = - ; cos x = 14. sec x = 5
41
4
26. B
sin x =

12
cos
1
11. -
6- 2
;
4
cos
3. C
9.
12
=

6+ 2
=
;
12
4

tan = 2 - 3
12
2. A
8.

5 41
41
; csc x =
5
41
25. F
117 4 117
quadrant II
; ;125 5
44
44 3 44
28.
; quadrant I
; ;
125 5 117
27.
15. Use the fact that 165º = 180º − 15º.
6- 2
- 6- 2
sin165 =
; cos165 =
;
4
4
tan165 = -2 + 3; cot165 = -2 - 3;
sec165 = - 6 + 2; csc165 = 6 + 2
29.
30.
2 + 3 7 2 3 + 21 2 + 3 7
;quadrant II
;
;
10
10
2 3 - 21
-2 30 + 2 5 - 4 6 -2 30 + 2
;
;
;
15
15
5+4 6
quadrant IV
Copyright © 2013 Pearson Education, Inc.
Chapter 5 Review Exercises
4 - 9 11 12 11 - 3 4 - 9 11
;
;
;
50
50
12 11 + 3
quadrant IV
231 - 2 -2 3 + 77
231 - 2
32.
;
;
;
18
18
-2 3 - 77
quadrant II
31.
33. sin  =
14
2
; cos  =
4
4
34. sin B = -
49. Verify sin 2 x - sin 2 y = cos 2 y - cos 2 x is an
identity.
(
= cos2 y - cos 2 x
cos 2 x - sin 2 x
is an
sec x
identity.
Work with the right side.
7
3
;cos B =
4
4
cos 2 x - sin 2 x cos 2 x - sin 2 x
=
1
sec x
cos x
(
)
= cos 2 x - sin 2 x ⋅ cos x
= cos3 x - sin 2 x cos x
(
)
= cos3 x - 1- cos 2 x cos x
= cos3 x - cos x + cos3 x
14
4
-1 + 5
39.
2
38.
= 2 cos3 x - cos x
51. Verify
6
3
sin 2 x
x
= cos 2 is an identity.
2 - 2 cos x
2
sin 2 x
1- cos 2 x
=
2 - 2 cos x 2 (1- cos x)
41. 0.5
42. -0.96
43. equivalent; -
sin 2 x + sin x
x
= cot
cos 2 x - cos x
2
44. equivalent;
1- cos 2 x
= tan x
sin 2 x
45. equivalent;
sin x
x
= cot
1- cos x
2
47. equivalent;
)
= 1- cos 2 x -1 + cos 2 y
1
37.
2
46. equivalent;
) (
sin 2 x - sin 2 y = 1- cos 2 x - 1- cos2 y
50. Verify 2 cos3 x - cos x =
4
3
35. cos 2 x = - ; sin 2 x =
5
5
24
7
36. sin 2 y = - ;cos 2 y = 25
25
40.
83
(
2 sin x - sin 3 x
cos x
) = sin 2 x
48. equivalent; csc x - cot x = tan
(1- cos x)(1 + cos x)
2 (1- cos x )
=
x
1 + cos x
= cos 2
2
2
sin 2 x
2
is an identity.
=
sin x
sec x
sin 2 x 2sin x cos x
=
= 2 cos x
sin x
sin x
2
2
=
=
1
sec x
cos x
52. Verify
cos x sin 2 x
= sin x
1 + cos 2 x
=
x
2
Copyright © 2013 Pearson Education, Inc.
84
Chapter 5 Trigonometric Identities
53. Verify 2 cos A - sec A = cos A -
tan A
is an
csc A
identity.
Work with the right side.
sin A
2
tan A
A = cos A - sin A
cos
cos A = cos A 1
csc A
cos A
sin A
=
=
2
2
2
2
2
2 1- tan 2 x
=
⋅
æ 2 tan x ö÷ tan x 2 tan x
ç
tan x ç
÷
çè1- tan 2 x ÷ø
=
1- tan 2 x
tan 2 x
1-
sin 2 x
2
cos2 x ⋅ cos x
sin 2 x cos 2 x
=
cos 2 x
cos A - sin A
cos A
(
2 cot x
= csc 2 x - 2 is an identity.
tan 2 x
2 cot x
=
tan 2 x
cos A sin A
cos A
cos A
2
=
56. Verify
=
2
)
cos A - 1- cos A
cos A
2 cos 2 A -1
1
=
= 2 cos A cos A
cos A
= 2 cos A - sec A
2 tan B
= sec2 B is an identity.
sin 2 B
sin B
2⋅
2 tan B
2sin B
cos B =
=
sin 2 B 2sin B cos B 2sin B cos 2 B
1
=
= sec 2 B
cos 2 B
cos 2 x - sin 2 x
= sec2 α = 1 + tan 2 α
1- 2sin 2 x
sin 2 x
2
= csc x - 2
57. Verify tan θ sin 2θ = 2 - 2 cos 2 θ is an
identity.
tan θ sin 2θ = tan θ (2sin θ cos θ )
=
54. Verify
55. Verify 1 + tan 2 α = 2 tan α csc 2α is an
identity.
Work with the right side.
sin α
2⋅
2 tan α
cos
α
=
2 tan α csc 2α =
sin 2α
2sin α cos α
2sin α
1
=
=
2
2sin α cos α cos 2 α
sin 2 x
=
sin θ
(2sin θ cos θ ) = 2sin 2 θ
cos θ
(
)
= 2 1- cos 2 θ = 2 - 2 cos 2 θ
58. Verify csc A sin 2 A - sec A = cos 2 A sec A is
an identity.
csc A sin 2 A - sec A
1
1
=
(2sin A cos A) sin A
cos A
= 2 cos A =
1
2 cos 2 A
1
=
cos A
cos A
cos A
2 cos 2 A -1 cos 2 A
=
= cos 2 A sec A
cos A
cos A
59. Verify 2 tan x csc 2 x - tan 2 x = 1 is an
identity.
2 tan x csc 2 x - tan 2 x
= 2 tan x
= 2⋅
=
1
- tan 2 x
sin 2 x
sin x
1
sin 2 x
⋅
cos x 2sin x cos x cos 2 x
1
cos 2 x
Copyright © 2013 Pearson Education, Inc.
-
sin 2 x
cos 2 x
=
1- sin 2 x
cos 2 x
=
cos 2 x
cos 2 x
=1
Chapter 5 Review Exercises
60. Verify 2 cos 2 θ -1 =
1- tan 2 θ
2
1 + tan θ
Work with the right side.
1-
1- tan 2 θ
=
1 + tan 2 θ
1+
=
.
=
sin 2 θ
=
cos 2 θ
=
2
cos 2 θ ⋅ cos θ
sin 2 θ cos 2 θ
cos 2 θ - sin 2 θ
cos 2 θ + sin 2 θ
=
cos 2 θ - sin 2 θ
1
= cos 2 θ - sin 2 θ
2
an identity.
Work with the right side.
2 tan θ cos 2 θ - tan θ
1- tan 2 θ
=
=
=
=
(
1 - tan θ
cos 2 α - sin 2 α + 1
(1- cos2 α ) + sin 2 α
cos 2 α + (1- sin 2 α )
sin 2 α + sin 2 α
cos 2 α + cos 2 α
is
(
=
)
(
)
64. Verify sin 3 θ = sin θ - cos 2 θ sin θ is an
identity.
Work with the right side.
(
= sin θ - sin θ + sin 3 θ
(
) = tan θ cos2 θ
= sin 3 θ
tan θ cos 2 θ 2 cos 2 θ -1
2 cos 2 θ -1
sec 2α -1
is an identity.
sec 2α + 1
Work with the right side.
sec 2α -1
sec 2α + 1
62. Verify sec2 α -1 =
2
cos α - sin α
)
sin θ - cos 2 θ sin θ = sin θ - 1- sin 2 θ sin θ
cos 2 θ - sin 2 θ
2
)
= 2sin 3 x - sin x
tan θ cos 2 θ 2 cos 2 θ -1
2
2
= cos α - sin α
1
2sin 2 x -1 sin x
⋅
1
sin x
sin x
(
tan θ 2 cos θ -1 cos 2 θ
⋅
sin 2 θ
cos 2 θ
1cos 2 θ
1
-1
= cos 2α
1
+1
cos 2α
1
)
= 2sin 2 x -1 sin x
2
(
2 cos 2 α
2
2
sin 2 x - cos 2 x sin x - 1- sin x
=
1
csc x
sin x
)
2
2sin 2 α
= tan 2 α = sec 2 α -1
tan θ 2 cos 2 θ -1
1- tan θ
=
sin 2 x - cos 2 x
= 2sin 3 x - sin x is an
csc x
identity.
= cos 2θ = 2 cos θ -1
2 tan θ cos 2 θ - tan θ
)
63. Verify
2
61. Verify tan θ cos 2 θ =
(
1- cos 2 α - sin 2 α
85
-1
⋅
65. Verify tan 4θ =
2 tan 2θ
is an identity.
2 - sec2 2θ
2 tan 2θ
tan 4θ = tan éë 2 (2θ )ùû =
1- tan 2 2θ
2 tan 2θ
2 tan 2θ
=
=
2
2 - sec2 2θ
1- sec 2θ -1
cos 2 α - sin 2 α
2
2
+ 1 cos α - sin α
Copyright © 2013 Pearson Education, Inc.
(
)
86
Chapter 5 Trigonometric Identities
x
tan x = tan x + sin x is an
2
identity. Work with the right side.
æ 1
ö
sin x
tan x + sin x =
+ sin x = sin x çç
+ 1÷
çè cos x ø÷÷
cos x
cos 2x
66. Verify 2 cos 2
æ
ö÷
çç
÷
1
ç
= sin x ç
+ 1÷÷
çç cos é 2 x ù ÷÷÷
êë ( 2 )úû ø
èç
=
=
( ) -1
2sin x cos 2 ( 2x )
2 cos 2 2x
cos x
=
cos 2
1
x 1
x
cot - tan = cot x is an identity.
2
2 2
2
1
x 1
x
cot - tan
2
2 2
2
1 æç1 + cos x ö÷ 1 æç1- cos x ÷ö
= ç
÷- ç
÷
2 çè sin x ÷ø 2 çè sin x ÷ø
1 + cos x 1- cos x
2sin x
2sin x
2 cos x
=
= cot x
2sin x
=
cos éê 2 ( 2x )ùú
ë
û
()
= 2 cos 2 2x tan x
4
x sin 2 x + sin x
is an identity.
=
2 cos 2 x - cos x
Work with the right side.
sin 2 x + sin x
2sin x cos x + sin x
=
cos 2 x - cos x
2 cos 2 x -1 - cos x
69. Verify - cot
2
Working with the right side, we have
1
sin x 1 + sin x
sec x + tan x =
+
=
cos x cos x
cos x
2 x
2 x
cos 2 + sin 2 + sin éê 2 ( 2x )ùú
ë
û
=
é
ù
x
cos ê 2 ( 2 )ú
ë
û
=
cos 2x
1- tan 2x
68. Verify
æx π ö
67. Verify tan çç + ÷÷÷ = sec x + tan x is an
çè 2 4 ø
identity. Working with the left side, we have
tan 2x + 1
æ x π ö tan 2x + tan π4
tan çç + ÷÷÷ =
=
çè 2 4 ø 1- tan x tan π 1- tan x
(
sin 2x
2
2sin x cos 2 ( 2x )
2
x
1 + tan 2x
the statement is verified.
)
2sin x cos 2 ( 2x )
cos 2x
=
æ x π ö tan 2x + 1
Since tan çç + ÷÷÷ =
= sec x + tan x ,
çè 2 4 ø 1- tan x
æ
ö÷
çç
1
÷
= sin x çç
+ 1÷÷
2
÷
x
ççè 2 cos ( 2 ) -1 ø÷÷
æ1 + 2 cos 2 x -1 ö÷
çç
( 2 ) ÷÷
÷
= sin x çç
çç 2 cos 2 ( x ) -1 ÷÷÷
çè
2
ø÷
(
=
sin 2x
+
cos 2x cos 2x
)
cos 2 2x + sin 2 2x + 2sin 2x cos 2x
cos 2 x - sin 2 x
2
(cos 2x + sin 2x )
=
(cos 2x + sin 2x )(cos 2x - sin 2x )
1
cos 2x + sin 2x cos 2x
=
⋅
1
cos 2x - sin 2x
cos 2x
Copyright © 2013 Pearson Education, Inc.
(
=
=
)
sin x (2 cos x + 1)
2 cos 2 x - cos x -1
sin x (2 cos x + 1)
(2 cos x + 1)(cos x -1)
sin x
sin x
=1- cos x
cos x -1
1
1
x
=== - cot
x
sin x
2
tan 2
cos x -1
=
Chapter 5 Test
5t
sin 3t + sin 2t tan 2
is an identity.
=
sin 3t - sin 2t
tan 2t
Using sum-to-product identities, we have
70. Verify
(
(
sin 3t + sin 2t 2sin
=
sin 3t - sin 2t 2 cos
=
) cos (
)sin (
3t -2t
2
3t -2t
2
)
)
sin 52t cos 2t
cos 52t sin 2t
= tan
71. (a) D =
3t +2t
2
3t +2t
2
tan 52t
5t
t
cot =
2
2 tan t
2
 2 sin 2
1
72. (a)

6- 2
4
4.
5. (a) - sin x
(b) tan x
6. -
2- 2
2
x
7. equivalent; cot - cot x = csc x
2
33
8. (a)
65
(b) -
32
(b) » 35 ft
63
16
(d) quadrant II
(b) V = a sin 2πω t and I = b sin 2πω t
W = VI = (a sin 2πωt )(b sin 2πωt )
= ab sin 2 2πωt
9. (a)
-
(b) -
cos 2 A = 1- 2sin 2 A 
cos 2 A -1 = -2sin 2 A 
1- cos 2 A
, so
2
W = ab sin 2 2πωt
= ab ⋅
7
25
24
25
(c)
24
7
(d)
5
5
2sin 2 A = 1- cos 2 A 
(e) 2
1- cos 2 (2πωt )
2
1- cos 4πωt
= ab ⋅
2
2π
2π
1 1
Thus, the period of V is
=
= ⋅ ,
b
4πω 2 ω
which is one-half the period of the voltage.
Additional answers will vary.
Chapter 5 Test
7
7
24
;tan  = - ;cot  = - ;
25
24
7
25
25
sec  = ;csc  = 24
7
1. sin  = -
56
65
(c)
.
sin 2 A =
87
10. Verify sec2 B =
1
is an identity.
1- sin 2 B
Work with the right side.
1
1
=
= sec2 B
2
1- sin B cos 2 B
cot A - tan A
is an identity.
csc A sec A
Work with the right side.
cos A sin A
cot A - tan A
sin A cos A
= sin A cos A ⋅
æ 1 öæ
csc A sec A
÷÷çç 1 ÷÷ö sin A cos A
çç
֍ cos A ֿ
èç sin A øè
11. Verify cos 2 A =
2. cos 
3. -1
Copyright © 2013 Pearson Education, Inc.
= cos 2 A - sin 2 A = cos 2 A
88
Chapter 5 Trigonometric Identities
æπ
ö
15. (a) V = 163cos çç - ω t ÷÷÷ .
çè 2
ø
sin 2 x
= tan x is an identity.
cos 2 x + 1
sin 2 x
2sin x cos x
=
cos 2 x + 1
2 cos 2 x -1 + 1
12. Verify
(
=
)
2sin x cos x
2
2 cos x
=
(b) 163 volts;
sin x
= tan x
cos x
2
13. Verify tan 2 x - sin 2 x = ( tan x sin x) is an
identity.
tan 2 x - sin 2 x =
=
=
sin 2 x
cos 2 x
- sin 2 x
sin 2 x - sin 2 x cos 2 x
(
cos 2 x
sin 2 x 1- cos 2 x
cos 2 x
) = sin 2 x sin 2 x
cos 2 x
2
= tan 2 x sin 2 x = ( tan x sin x )
tan x - cot x
= 2sin 2 x -1 is an
tan x + cot x
identity.
sin x cos x
tan x - cot x cos x sin x
=
tan x + cot x sin x + cos x
cos x sin x
sin x cos x
x sin x ⋅ cos x sin x
cos
=
sin x cos x cos x sin x
+
cos x sin x
14. Verify
=
sin 2 x - cos 2 x
sin 2 + cos 2 x
= sin 2 x - cos 2 x
(
= sin 2 x - 1- sin 2 x
)
= 2sin 2 x -1
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sec
240