Product-to-sum and Sum-to-Product identities
Sum-to-Product identities
Here are the sum-to-product identities: \begin{align*} \sin (x) + \sin (y) &= 2 \sin \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right) \\ \sin (x) - \sin (y) &= 2 \sin \left(\frac{x - y}{2}\right) \cos \left(\frac{x + y}{2}\right) \\ \cos (x) + \cos (y) &= 2 \cos \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right) \\ \cos (x) - \cos (y) &= -2 \sin \left(\frac{x + y}{2}\right) \sin \left(\frac{x - y}{2}\right) \end{align*}
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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