Identities

Math 335
Trigonometric Identities
Reciprocal Identities
1
sin u = csc u
1
cos u = sec u
1
tan u = cot u
1
csc u = sin u
1
sec u = cos u
1
cot u = tan u
Quotient Identities
sin u
tan u = cos u
cos u
cot u = sin u
Pythagorean Identities
sin2 u + cos2 u = 1
2
1 + tan u = sec2 u
1 + cot2 u = csc2 u
Cofunction Identities
sin ( 90° – u ) = cos u
π
tan ( 2 – u ) = cot u
sec ( 90° – u ) = csc u
cos ( 90° – u ) = sin u
π
cot ( 2 – u ) = tan u
π
csc ( 2 – u ) = sec u
Even/Odd Identities
sin ( – u ) = – sin u
cos ( – u ) = cos u
tan ( – u ) = – tan u
csc ( – u ) = – csc u
sec ( – u ) = sec u
cot ( – u ) = – cot u
Sum and Difference Identities
cos ( u – v ) = cos u cos v + sin u sin v
cos ( u + v ) = cos u cos v – sin u sin v
sin ( u + v ) = sin u cos v + cos u sin v
sin ( u – v ) = sin u cos v – cos u sin v
tan u + tan v
tan ( u + v ) = 1 – tan u tan v
tan ( u – v ) =
tan u – tan v
1 + tan u tan v
Double-Angle Identities
sin 2u = 2 sin u cos u
cos 2u = cos2 u – sin2 u
= 2 cos2 u – 1
= 1 – 2 sin2 u
tan 2u =
2 tan u
1 – tan2u
Half-Angle Identities
u
sin 2 = ±
1 – cos u
2
u
cos 2 = ±
1 + cos u
2
u
tan 2 = ±
1 – cos u
1 + cos u
=
1 – cos u
sin u
=
sin u
1 + cos u