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