\textbf{Double angle formulas:} 

By setting \[\beta = \alpha\] we get double angle formulas. There is one double-angle formula for sine but three useful double angle formulas for cosine. The three are easily from each other using the fundamental identity \[\sin^2(\alpha) + \cos^2(\alpha) = 1\]

,,\qquad \sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)
\[\qquad\begin{aligned} \cos(2\alpha) &= \cos^2(\alpha) - \sin^2(\alpha)\\&= 1 - 2\sin^2(\alpha)\\&= 2\cos^2(\alpha) - 1\end{aligned}\]

In additon, double angle formulas can be expressed in terms of a tangent:

,,\qquad \sin(2\alpha) = \frac{2\tan(\alpha)}{1 + \tan^2(\alpha)}
,,\qquad \cos(2\alpha) = \frac{2\tan(\alpha)}{1 + \tan^2(\alpha)}
,,\qquad \tan(2\alpha) = \frac{2\tan\alpha}{1 - \tan^2\alpha}