Pythagorean Identities
\cos^2 x + \sin^2 x = 1
\sec^2 x - \tan^2 x = 1
Double-Angle Identities
\sin 2x = 2 \sin x \cos x
\cos 2x = \cos^2 x - \sin^2 x = 1 - 2 \sin^2 x = 2 \cos^2 x - 1
Half-Angle Identities
\cos^2 x = \dfrac{1+ \cos 2x}{2}
\sin^2 x = \dfrac{1- \cos 2x}{2}
Angle Sum and Difference Identities
\sin(α + β) = \sin(α) \cos(β) + \cos(α) \sin(β)
\sin(α - β) = \sin(α) \cos(β) - \cos(α) \sin(β)
\cos(α + β) = \cos(α) \cos(β) - \sin(α) \sin(β)
\cos(α - β) = \cos(α) \cos(β) + \sin(α) \sin(β)
Angle Reflections and Shifts
\sin (-x) = -\sin x
\cos(-x) = \cos x
\tan (-x) = -\tan x
\sin\left(x \pm \frac{\pi}{2}\right) = \pm \cos x
\cos\left(x \pm \frac{\pi}{2}\right) = \mp \sin x
Angle Supplement Identities
\sin(\pi - x) = \sin x
\cos(\pi - x) = -\cos x
\tan(\pi - x) = -\tan x
Periodicity Identities
\sin(x+2\pi) = \sin x
\cos(x+2\pi) = \cos x
\tan(x+\pi) = \tan x
