Trigonometric Identities
In calculus, trigonometric functions come up often and in a variety of contexts. The trigonometric identities listed below are important tools when looking for ways to make it easier to differentiate or integrate an otherwise more complex function. Substitute in these trigonometric identities when needed to find different ways to differentiate or integrate.
Reciprocal Identities
sin x = 1/csc x
cos x = 1/sec x
tan x = 1/cot x
csc x = 1/sin x
sec x = 1/cos x
cot x = 1/tan x
Pythagorean Identities
sin2 x + cos2 x = 1
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
Quotient Identities
tan x = sin x/cos x
cot x = cos x/sin x
Co-Function Identities
sin (Π/2 - x) = cos x
cos (Π/2 - x) = sin x
tan (Π/2 - x) = cot x
csc (Π/2 - x) = sec x
sec (Π/2 - x) = csc x
cot (Π/2 - x) = tan x
Even-Odd Identities
sin (-x) = -sin x
cos (-x) = cos x
tan (-x) = -tan x
csc (-x) = -csc x
sec (-x) = sec x
cot (-x) = -cot x
Sum-Difference Formulas
sin (u ± v) = sin u cos v ± cos u sin v
cos (u ± v) = cos u cos v ∓ sin u sin v
tan (u ± v) = (tan u ± tan v)/(1 ∓ tan u tan v)