# Fundamental Trigonometric Identities

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## "arc" Identities

$\arctan\theta=\tan^{-1}\theta$

$\arcsin\theta=\sin^{-1}\theta$

$\arccos\theta=\cos^{-1}\theta$

## Quotient and reciprocal identities

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$

$\cot\theta=\dfrac{\cos\theta}{\sin\theta}= \dfrac{\csc\theta}{\sec\theta}= \dfrac{1}{\tan\theta}$

$\sec\theta=\dfrac{1}{\cos\theta}$

$\csc\theta=\dfrac{1}{\sin\theta}$

## Cofunction Function identities

$\sin\theta = \cos(\dfrac{\pi}{2} - \theta)$

$\cos\theta = \sin(\dfrac{\pi}{2} - \theta)$

## Even/Odd Functions

$\cos(-\theta) = \cos(\theta)$

$\sin(-\theta) = -\sin(\theta)$

## Pythagorean identities

$\sin^2\theta+\cos^2\theta=1$

$\tan^2\theta+1=\sec^2\theta$

$1+\cot^2\theta=\csc^2\theta$

## Angle sum and difference identities

$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$

$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\sin\beta\cos\alpha$

$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$

$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$

$\tan(\alpha+\beta) = \dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$

$\tan(\alpha-\beta) = \dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$

## Double-angle identities

$\sin2\theta=2\sin\theta\cos\theta$

$\cos2\theta=\cos^2\theta-\sin^2\theta = 2\cos^2\theta-1 = 1-2\sin^2\theta$

$\tan2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}$

## Half-angle identities

$\sin\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1-\cos\theta}{2}}$

$\cos\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1+\cos\theta}{2}}$

$\tan\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1-\cos\theta}{1+\cos\theta}} = \dfrac{\sin\theta}{1+\cos\theta} = \dfrac{1-\cos\theta}{\sin\theta}$

## Reduction formulas

$\sin^2\theta=\dfrac{1-\cos2\theta}{2}$

$\cos^2\theta=\dfrac{1+\cos2\theta}{2}$

$\tan^2\theta=\dfrac{1-\cos2\theta}{1+\cos2\theta} = \dfrac{\sin2\theta}{1+\cos2\theta} = \dfrac{1-\cos2\theta}{\sin2\theta}$

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