
# 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

$$\sin (-x) = -\sin x$$

$$\cos(-x) = \cos x$$

$$\sin\left(x \pm \frac{\pi}{2}\right) = \pm \cos x$$

$$\cos\left(x \pm \frac{\pi}{2}\right) = \mp \sin x$$

Trigonometric Identities is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Paul Seeburger.