Universal Trigonometric Substitution

Theorem: Universal Trigonometric Substitution

The sine, cosine, tangent and cotangent of $\theta$ can all be expressed in terms of $\tan \frac{\theta }{2}$:

$\sin \theta =\frac{2\tan \frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}$

$\cos \theta =\frac{1-{\tan }^2\frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}$

$\tan \theta =\frac{2\tan \frac{\theta }{2}}{1-{\tan }^2\frac{\theta }{2}}$

$\cot \theta =\frac{1-{\tan }^2\frac{\theta }{2}}{2\tan \frac{\theta }{2}}$

PROOF

TODO