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*}