\begin{equation*}
\begin{aligned}
\sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\\
\cos (\alpha+\beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
\end{aligned}
\end{equation*}
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\begin{equation*}
\begin{aligned}
\sin (\alpha-\beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\\
\cos (\alpha-\beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
\end{aligned}
\end{equation*}
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