Product-to-sum identities

The product-to-sum identities are as follows:

$\sin (x) \sin (y) = \frac{1}{2} (\cos (x-y) - \cos (x+y))$ $\sin (x) \cos (y) = \frac{1}{2} (\sin (x-y) + \sin (x+y))$ $\cos (x) \cos (y) = \frac{1}{2} (\cos (x-y) + \cos (x+y))$ They can be derived by expanding out $\cos (x+y)$ and $\cos (x-y)$ or $\sin (x+y)$ and $\sin(x-y)$ , then combining them to isolate each term.