## Trig Formulas and Identities

## Handy Table of Trig Formulas and Identities

### Relationships Among Trig Functions

\begin{align*}

\csc x &= \frac{1}{\sin x} & \sec x &= \frac{1}{\cos x} \\ \\

\tan x &= \frac{\sin x}{\cos x} & \cot x &= \frac{\cos x}{\sin x}

\end{align*}

\begin{align*}

\sin^2 x + \cos^2 x &= 1 \\

1 + \cot^2 x &= \csc^2 x \\

1 + \tan^2 x &= \sec^2 x

\end{align*}

### Negative Angles

\begin{align*}

\sin(-x) &= -\sin x & \csc(-x) &= -\csc x\\

\cos(-x) &= \cos x & \sec(-x) &= \sec x \\

\tan(-x) &= -\tan x & \cot(-x) &= -\cot x

\end{align*}

### Addition and Subtraction Formulas

\begin{align*}

\sin(x+y) &= \sin x \cos y + \sin y \cos x \\ \\

\sin(x – y) &= \sin x \cos y – \sin y \cos x \\ \\

\cos(x+y) &= \cos x \cos y – \sin x \sin y \\ \\

\cos(x – y) &= \cos x \cos y + \sin x \sin y \\ \\

\tan(x+y) &= \frac{\tan x + \tan y}{1 – \tan x \tan y} \\ \\

\tan(x-y) &= \frac{\tan x – \tan y}{1 + \tan x \tan y}

\end{align*}

### Double-Angle Formulas

\begin{align*}

\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 \\ \\

\tan 2x &= \frac{2\tan x}{1 – \tan^2 x}

\end{align*}

### Half-Angle Formulas

(useful for $\sin^2 x$ & $\cos^2 x$)

\begin{align*}

\sin^2 x &= \frac{1}{2} – \frac{1}{2}\cos 2x \\ \\

\cos^2 x &= \frac{1}{2} + \frac{1}{2}\cos 2x

\end{align*}