cos sin 1 1 tan sec cot 1 csc x x x x x x + = + = + =

Basic Identities
csc x =
1
sin x
1
sec x =
cos x
cot x =
tan x =
sin x =
1
csc x
1
cos x =
sec x
1
tan x
tan x =
sin x
cos x
cot x =
1
cot x
cos x
sin x
Even/Odd Identities
Cofunction Identities
Double Angle Identities
π

sin  − θ  = cos θ
2

π

cos  − θ  = sin θ
2

sin 2u = 2 sin u cos u
sin u sin v =
cos 2u = cos 2 u − sin 2 u
cos u cos v =
1
2
[cos(u − v ) + cos(u + v )]
= 1 − 2 sin 2 u
sin u cos v =
1
2
[sin(u + v ) + sin (u − v )]
2 tan u
1 − tan 2 u
cos u sin v =
1
2
[sin(u + v ) − sin (u − v )]
π

tan  − θ  = cot θ
2

π

cot  − θ  = tan θ
2

π

sec  − θ  = csc θ
2

π

csc  − θ  = sec θ
2

[cos(u − v ) − cos(u + v )]
= 2 cos u − 1
tan 2u =
Power-Reducing Identities
sin 2 u =
1 − cos 2u
2
Sum-to-Product Identities
Sum and Difference Identities
tan ( − x ) = − tan x cot ( − x ) = − cot x
1
2
2
cos ( − x ) = cos x sec ( − x ) = sec x
sin ( − x ) = − sin x csc ( − x ) = − csc x
Product-to-Sum Identities
cos 2 u =
1 + cos 2u
2
u+v
u −v
sin u + sin v = 2sin 
 cos 

 2 
 2 
tan 2 u =
1 − cos 2u
1 + cos 2u
u +v  u −v 
sin u − sin v = 2 cos 
 sin 

 2   2 
sin (u + v ) = sin u cos v + cos u sin v
sin (u − v ) = sin u cos v − cos u sin v
Half-Angle Identities
Pythagorean Identities
cos 2 x + sin 2 x = 1
2
sin
cos(u − v ) = cos u cos v + sin u sin v
1 − cos u
u
=±
2
2
tan (u + v ) =
tan u + tan v
1 − tan u tan v
cos
2
1 + cos u
u
=±
2
2
2
tan (u − v ) =
tan u − tan v
1 + tan u tan v
tan
u 1 − cos u
sin u
=
=
2
sin u
1 + cos u
1 + tan x = sec x
2
cos(u + v ) = cos u cos v − sin u sin v
cot x + 1 = csc x
u+v
u −v 
cos u + cos v = 2 cos 
 cos 

 2 
 2 
u +v u −v
cos u − cos v = −2sin 
 sin 

 2   2 