Double-Angle Formulas

\begin{align} \sin 2\alpha &=2\sin \alpha\cos \alpha= \frac{2\tan \alpha }{1 + \tan^{2} \alpha} \\ \\ \cos 2\alpha &=\cos^{2} \alpha -\sin^{2} \alpha= \frac{1-\tan^{2} \alpha }{1 + \tan^{2} \alpha} \\ \\ \tan 2\alpha &=\frac{2\tan \alpha }{1 - \tan^{2} \alpha} \\ \\ \cot 2\alpha &=\frac{\cot^{2} \alpha -1 }{2\cot \alpha } \\ \end{align}