# 14.3.3: Trigonometric Identities

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## 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

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

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

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

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

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