Trigonometric Identities, Thrice

Trigonometric Identities, Thrice
Prove each identity.
1.
cos(270° − 𝛽) = − sin 𝛽
2.
cos ( − 𝛽) =
3.
sin(45° + 𝛼) =
√2
(cos 𝛼
2
4.
tan(45° − 𝛼) =
1−tan 𝛼
1+tan 𝛼
5.
tan ( 4 + 𝛼) = tan 𝛼+1
6.
tan(360° − 𝛽) = − tan 𝛽
7.
tan(𝛽 + 45°) + tan(𝛽 − 45°) = 2 tan 2𝛽
8.
cos 2𝛼 = cos2 𝛼 − sin2 𝛼
9.
2 csc 𝜆 = csc 𝜆−cot 𝜆 + csc 𝜆+cot 𝜆
10.
1−sin 𝛼
1+sin 𝛼
11.
(sin 𝜃 + cos 𝜃)2 tan 𝜃 = tan 𝜃 + 2 sin2 𝜃
12.
(1 + sin 𝜃 + cos 𝜃)(1 − sin 𝜃 − cos 𝜃) = −2 sin 𝜃 cos 𝜃
13.
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) =
𝜋
3
3𝜋
cos 𝛽+√3 sin 𝛽
2
+ sin 𝛼)
tan 𝛼−1
1
1
= (tan 𝛼 − sec 𝛼)2
sec2 𝜃−tan2 𝜃+tan 𝜃
cos 𝜃
Trigonometric Identities, Thrice KEY
Prove each identity.
1.
cos(270° − 𝛽) = − sin 𝛽
2.
cos 270° cos 𝛽 + sin 270° sin 𝛽 = − sin 𝛽
𝜋
3
cos ( − 𝛽) =
cos 𝛽+√3 sin 𝛽
2
𝜋
𝜋
cos 𝛽 + √3 sin 𝛽
cos cos 𝛽 + sin sin 𝛽 =
3
3
2
(0) cos 𝛽 + (−1) sin 𝛽 = − sin 𝛽
1
cos 𝛽 + √3 sin 𝛽
√3
( ) cos 𝛽 + ( ) sin 𝛽 =
2
2
2
− sin 𝛽 = − sin 𝛽
cos 𝛽 + √3 sin 𝛽 cos 𝛽 + √3 sin 𝛽
=
2
2
3.
sin(45° + 𝛼) =
√2
(cos 𝛼
2
4.
+ sin 𝛼)
sin 45° cos 𝛼 + cos 45° sin 𝛼 =
√2
(cos 𝛼
2
tan 45°−tan 𝛼
1+tan 45° tan 𝛼
+
sin 𝛼)
(
√2
√2
√2
(cos 𝛼 + sin 𝛼)
) cos 𝛼 + ( ) sin 𝛼 =
2
2
2
√2
(cos 𝛼
2
5.
+ sin 𝛼) =
3𝜋
√2
(cos 𝛼
2
tan 𝛼−1
tan ( 4 + 𝛼) = tan 𝛼+1
3𝜋
+tan 𝛼
4
3𝜋
1−tan tan 𝛼
4
tan
−1+tan 𝛼
1−(−1) tan 𝛼
=
=
tan 𝛼−1
tan 𝛼+1
tan 𝛼−1
tan 𝛼+1
tan(45° − 𝛼) =
=
1−tan 𝛼
1+tan 𝛼
1−tan 𝛼
1+tan 𝛼
1−tan 𝛼
1+(1) tan 𝛼
= 1+tan 𝛼
1−tan 𝛼
1−tan 𝛼
1+tan 𝛼
1−tan 𝛼
= 1+tan 𝛼
+ sin 𝛼)
6.
tan(360° − 𝛽) = − tan 𝛽
tan 360°+tan 𝛽
1−tan 360° tan 𝛽
0+tan 𝛽
1−(0) tan 𝛽
= − tan 𝛽
= − tan 𝛽
− tan 𝛽 = − tan 𝛽
tan 𝛼−1
tan 𝛼+1
=
tan 𝛼−1
tan 𝛼+1
7.
tan(𝛽 + 45°) + tan(𝛽 − 45°) = 2 tan 2𝛽
tan 𝛽+tan 45°
tan 𝛽−tan 45°
+
1−tan 𝛽 tan 45°
1+tan 𝛽 tan 45°
tan 𝛽+1
tan 𝛽−1
+ 1+tan 𝛽(1)
1−tan 𝛽(1)
tan 𝛽+1 1+tan 𝛽
∙
1−tan 𝛽 1+tan 𝛽
cos 𝛼 cos 𝛼 − sin 𝛼 sin 𝛼 = cos 2 𝛼 − sin2 𝛼
= 2 tan 2𝛽
cos2 𝛼 − sin2 𝛼 = cos2 𝛼 − sin2 𝛼
tan 𝛽−1 1−tan 𝛽
+ 1+tan 𝛽 ∙ 1−tan 𝛽 = 2 tan 2𝛽
= 2 tan 2𝛽
= 2 tan 2𝛽
2 tan 𝛽
)
1−tan2 𝛽
2(
= 2 tan 2𝛽
tan 𝛽+tan 𝛽
)
1−tan 𝛽 tan 𝛽
2(
= 2 tan 2𝛽
2 tan(𝛽 + 𝛽) = 2 tan 2𝛽
2 tan 2𝛽 = 2 tan 2𝛽
9.
1
1
2 csc 𝜆 = csc 𝜆−cot 𝜆 + csc 𝜆+cot 𝜆
2 csc 𝜆 =
1
csc 𝜆+cot 𝜆
∙
csc 𝜆−cot 𝜆 csc 𝜆+cot 𝜆
2 csc 𝜆 =
csc 𝜆+cot 𝜆+csc 𝜆−cot 𝜆
csc2 𝜆−cot2 𝜆
2 csc 𝜆 =
2 csc 𝜆
csc2 𝜆−cot2 𝜆
2 csc 𝜆 =
2 csc 𝜆
1
2 csc 𝜆 = 2 csc 𝜆
cos 2𝛼 = cos2 𝛼 − sin2 𝛼
cos(𝛼 + 𝛼) = cos2 𝛼 − sin2 𝛼
= 2 tan 2𝛽
tan2 𝛽+2 tan 𝛽+1−tan2 𝛽+2tan 𝛽−1
1−tan2 𝛽
4 tan 𝛽
1−tan2 𝛽
8.
+
1
csc 𝜆+cot 𝜆
∙
csc 𝜆−cot 𝜆
csc 𝜆−cot 𝜆
10.
11.
1−sin 𝛼
1+sin 𝛼
= (tan 𝛼 − sec 𝛼)2
1−sin 𝛼
1+sin 𝛼
= tan2 𝛼 − 2 tan 𝛼 sec 𝛼 + sec 2 𝛼
1−sin 𝛼
1+sin 𝛼
= cos2 𝛼 − 1 cos 𝛼 cos 𝛼 + cos2 𝛼
1−sin 𝛼
1+sin 𝛼
=
sin2 𝛼−2 sin 𝛼+1
cos2 𝛼
1−sin 𝛼
1+sin 𝛼
=
sin2 𝛼−2 sin 𝛼+1
1−sin2 𝛼
1−sin 𝛼
1+sin 𝛼
=
1−2 sin 𝛼+sin2 𝛼
1−sin2 𝛼
1−sin 𝛼
1+sin 𝛼
= (1−sin
1−sin 𝛼
1+sin 𝛼
= 1+sin 𝛼
sin2 𝛼
2 sin 𝛼
1
1
(1−sin 𝛼)(1−sin 𝛼)
𝛼)(1+sin 𝛼)
1−sin 𝛼
(sin 𝜃 + cos 𝜃)2 tan 𝜃 = tan 𝜃 + 2 sin2 𝜃
(sin2 𝜃 + 2 sin 𝜃 cos 𝜃 + cos 2 𝜃) tan 𝜃 = tan 𝜃 + 2 sin2 𝜃
(1 + 2 sin 𝜃 cos 𝜃) tan 𝜃 = tan 𝜃 + 2 sin2 𝜃
tan 𝜃 + 2 sin 𝜃 cos 𝜃 tan 𝜃 = tan 𝜃 + 2 sin2 𝜃
2 sin 𝜃 cos 𝜃 sin 𝜃
1
1 cos 𝜃
tan 𝜃 + 1
= tan 𝜃 + 2 sin2 𝜃
tan 𝜃 + 2 sin2 𝜃 = tan 𝜃 + 2 sin2 𝜃
12.
(1 + sin 𝜃 + cos 𝜃)(1 − sin 𝜃 − cos 𝜃) = −2 sin 𝜃 cos 𝜃
1 − sin2 𝜃 − cos2 𝜃 − 2 sin 𝜃 cos 𝜃 = −2 sin 𝜃 cos 𝜃
cos2 𝜃 − cos 2 𝜃 − 2 sin 𝜃 cos 𝜃 = −2 sin 𝜃 cos 𝜃
−2 sin 𝜃 cos 𝜃 = −2 sin 𝜃 cos 𝜃
13.
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) =
sec2 𝜃−tan2 𝜃+tan 𝜃
cos 𝜃
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) =
1+tan 𝜃
cos 𝜃
1
tan 𝜃
1
sin 𝜃
cos 𝜃
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) = cos 𝜃 + cos 𝜃
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) = cos 𝜃 +
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) =
1
cos 𝜃
+
cos 𝜃
sin 𝜃
cos2 𝜃
1
sin 𝜃
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) = cos 𝜃 (1 + cos 𝜃)
1
sin 𝜃
1
sin 𝜃 1
1 cos 𝜃
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) = cos 𝜃 (cos 𝜃 + 1)
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) = cos 𝜃 (
+ 1)
sec 𝜃 (sin 𝜃 sec 𝜃 + 1) = sec 𝜃 (sin 𝜃 sec 𝜃 + 1)