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Trigonometric Identities

Fundamental Formulae

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

Double Angle Formulae

\sin(2\theta)=2\sin(\theta)\cos(\theta)
\cos(2\theta)=\cos(\theta)^2-\sin^2(\theta)
\cos(2\theta)=2\cos^2(\theta)-1
\cos(2\theta)=1-2\sin^2(\theta)

Compound Angle Formulae

\sin(\alpha\pm\beta)=\sin(\alpha)\cos(\beta)\pm\cos(\alpha)\sin(\beta)
\cos(\alpha\pm\beta)=\sin(\alpha)\sin(\beta)\mp\cos(\alpha)\cos(\beta)
\tan(\alpha\pm\beta)=\frac{\tan(\alpha)\pm\tan(\beta)}{1\mp\tan(\alpha)\tan(\beta)}

Example 1

Simplify

\frac{\sqrt{1-\cos^2(x)}}{\cos(x)}


Example 2

Simplify

\frac{\tan(\theta)}{\sec^2(\theta)-2}