Sine:
\begin{eqnarray*} \sin(2a)&=&\sin(a+a)\\ &=&\sin(a)\cos(a)+\cos(a)\sin(a)\\ &=&2\sin(a)\cos(a). \end{eqnarray*} Cosine: \begin{eqnarray*} \cos(2a)&=&\cos(a+a)\\ &=&\cos(a)\cos(a)+\sin(a)\sin(a)\\ &=&\cos^2(a)-\sin^2(a). \end{eqnarray*} By using the identity $$\sin^2(a)+\cos^2(a)=1$$ we can change the expression above into the alternate forms $$\cos(2a)=2\cos^2(a)-1 = 1-2\sin^2(a).$$

Tangent: \begin{eqnarray*} \tan(2a)&=&\tan(a+a)\\ &=&\frac{\tan(a)+\tan(a)}{1-\tan(a)\tan(a)}\\ &=&\frac{2\tan(a)}{1-\tan^2(a)}. \end{eqnarray*}