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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)}

Using the identity

\sin^2(x)+\cos^2(x)=1

we have

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

and so

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

Hence,

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


Example 2: Simplify

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

This can be rewritten as

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

Multiplying the top and bottom of the fraction by \cos^2(x) gives

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

Using the double angle formulae, this becomes

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