Trigonometric Identities

Pythagorean Identities

$\sin^2(\theta)+\cos^2(\theta)=1$
$1+\tan^2(\theta)=\sec^2(\theta)$
$1+\cot^2(\theta)=\csc^2(\theta)$

Reciprocal Identities

$\sin(\theta)=\frac{1}{\csc(\theta)}$
$\cos(\theta)=\frac{1}{\sec(\theta)}$
$\tan(\theta)=\frac{1}{\cot(\theta)}$
$\csc(\theta)=\frac{1}{\sin(\theta)}$
$\sec(\theta)=\frac{1}{\cos(\theta)}$
$\cot(\theta)=\frac{1}{\tan(\theta)}$

Power Reduction Identities

$\sin^2(\theta)=\frac{1-\cos(2\theta)}{2}$
$\cos^2(\theta)=\frac{1+\cos(2\theta)}{2}$
$\tan^2(\theta)=\frac{1-\cos(2\theta)}{1+\cos(2\theta)}$

Quotient Identities

$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$
$\cot(\theta)=\frac{\cos(\theta)}{\sin(\theta)}$

Law of Sines

$\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$

Law of Cosines

$a^2=b^2+c^2-2bc\cos(A)$
$b^2=a^2+c^2-2ac\cos(B)$
$c^2=a^2+b^2-2ab\cos(C)$

Half Angle Identities

$\sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos(\theta)}{2}}$
$\cos\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1+\cos(\theta)}{2}}$
$\tan\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos(\theta)}{1+\cos(\theta)}}$

Note that the sign depends on the quadrant of $\frac{\theta}{2}$ .

Double Angle Identities

$\sin(2\theta)=2\sin(\theta)\cos(\theta)$
$\cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)$
$\cos(2\theta)=2\cos^2(\theta)-1$
$\cos(2\theta)=1-2\sin^2(\theta)$
$\tan(2\theta)=\frac{2\tan(\theta)}{1-\tan^2(\theta)}$

Triple Angle Identities

$\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)$
$\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$
$\tan(3\theta) = \frac{3\tan(\theta) - \tan^3(\theta)}{1 - 3\tan^2(\theta)}$

Euler's Formula

$e^{i\theta} = \cos(\theta) + i\sin(\theta)$
$\sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}$
$\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}$

Even/Odd Identities

$\sin(-\theta)=-\sin(\theta)$
$\cos(-\theta)=\cos(\theta)$
$\tan(-\theta)=-\tan(\theta)$
$\csc(-\theta)=-\csc(\theta)$
$\sec(-\theta)=\sec(\theta)$
$\cot(-\theta)=-\cot(\theta)$

Cofunction Identities

$\sin\left(\frac{\pi}{2}-\theta\right)=\cos(\theta)$
$\cos\left(\frac{\pi}{2}-\theta\right)=\sin(\theta)$
$\tan\left(\frac{\pi}{2}-\theta\right)=\cot(\theta)$
$\csc\left(\frac{\pi}{2}-\theta\right)=\sec(\theta)$
$\sec\left(\frac{\pi}{2}-\theta\right)=\csc(\theta)$
$\cot\left(\frac{\pi}{2}-\theta\right)=\tan(\theta)$

Sum and Difference Identities

$\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$
$\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$
$\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$
$\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$
$\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}$
$\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}$

Sum-to-Product Identities

$\sin(a)+\sin(b)=2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)$
$\sin(a)-\sin(b)=2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)$
$\cos(a)+\cos(b)=2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)$
$\cos(a)-\cos(b)=-2\sin\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)$

Product-to-Sum Identities

$\sin(a)\sin(b)=\frac{1}{2}[\cos(a-b)-\cos(a+b)]$
$\cos(a)\cos(b)=\frac{1}{2}[\cos(a-b)+\cos(a+b)]$
$\sin(a)\cos(b)=\frac{1}{2}[\sin(a+b)+\sin(a-b)]$
$\cos(a)\sin(b)=\frac{1}{2}[\sin(a+b)-\sin(a-b)]$