# 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

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

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

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

Trigonometric Identities is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.