8.1: Trigonometric Identities

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

$\sin(x+2\pi) = \sin x$

$\cos(x+2\pi) = \cos x$

$\tan(x+\pi) = \tan x$