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Double-Angle and Half-Angle Identities

Double-Angle Identities

The Double-Angle Identities (these are really just special cases of Bhaskaracharya's formulas , when $u$ = $v$ )

$\begin{array}{l}\sin \left(2u\right)=2\sin \left(u\right)\cos \left(u\right)\\ \cos \left(2u\right)={\cos }^2\left(u\right)-{\sin }^2\left(u\right)\\ \cos \left(2u\right)=2{\cos }^2\left(u\right)-1\\ \cos \left(2u\right)=1-2{\sin }^2\left(u\right)\\ \tan \left(2u\right)=\frac{2\tan \left(u\right)}{1-{\tan }^2\left(u\right)}\end{array}$

Power-Reducing Identities

These can be derived from the identities above, by solving for ${\sin }^2\left(u\right)$ , ${\cos }^2\left(u\right)$ , or ${\tan }^2\left(u\right)$ .

$\begin{array}{l}{\sin }^2\left(u\right)=\frac{1-\cos \left(2u\right)}{2}\\ {\cos }^2\left(u\right)=\frac{1+\cos \left(2u\right)}{2}\\ {\tan }^2\left(u\right)=\frac{1-\cos \left(2u\right)}{1+\cos \left(2u\right)}\end{array}$

Half-Angle Identities

These are the same as the identities above, but with the square root of both sides taken, and $\theta$ substituted for $2u$ .

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