\begin{eqnarray*}
2\sin\alpha\cos\beta & = & \sin\left(\alpha+\beta\right)+\sin\left(\alpha-\beta\right)\\
2\cos\alpha\sin\beta & = & \sin\left(\alpha+\beta\right)-\sin\left(\alpha-\beta\right)\\
2\cos\alpha\cos\beta & = & \cos\left(\alpha+\beta\right)+\cos\left(\alpha-\beta\right)\\
2\sin\alpha\sin\beta & = & \cos\left(\alpha-\beta\right)-\cos\left(\alpha+\beta\right)
\end{eqnarray*}