How to Derive a Double Angle Identity. How does one derive the following two identities:
$$\begin{align*}
\cos 2\theta &= 1-2\sin^2\theta\\
\sin 2\theta &= 2\sin\theta\cos\theta
\end{align*}$$

The first one follows from:
$$\cos 2\theta = \cos(\theta +\theta) = \cos \theta \cos \theta - \sin \theta \sin \theta = \cos^2 \theta - \sin^2 \theta .$$ Now use the fact $\cos^2 \theta + \sin^2 \theta =1.$  
The second one follows from 
$$ \sin 2\theta = \sin(\theta +\theta) = \sin \theta \cos \theta + \cos \theta \sin \theta. $$
