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 identitie s

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

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

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

Reduction formulas

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

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

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