Product-to-sum identities
The product-to-sum identities are as follows:
\begin{align*}
\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))
\end{align*}
They can be derived by expanding out 
and 
or 
and 
, then combining them to isolate each term.
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