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\subsection*{Product to sum}

\begin{align*}
  \sin \theta \sin y &= \frac{1}{2}\big[\cos(\theta - y) - \cos(\theta + y)\big]\\
  \cos \theta \cos y &= \frac{1}{2}\big[\cos(\theta - y) + \cos(\theta + y)\big]\\
  \sin \theta \cos y &= \frac{1}{2}\big[\sin(\theta + y) + \sin(\theta - y)\big]\\
  \tan \theta \tan y &= \frac{ \tan \theta + \tan y }{ \cot \theta + \cot y }\\
  \tan \theta \cot y &= \frac{ \tan \theta + \cot y }{ \cot \theta + \tan y }
\end{align*}