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Trigonometric identities

\begin{align} \sin(x+y)&=\sin x\cos y + \cos x\sin y \label{eq:sinplus}\\ \sin(x-y)&=\sin x\cos y - \cos x\sin y \label{eq:sinminus}\\ \cos(x+y)&=\cos x\cos y - \sin x\sin y \label{eq:cosplus}\\ \cos(x-y)&=\cos x\cos y + \sin x\sin y \label{eq:cosminus} \end{align}

From Eqs. \eqref{eq:sinplus}-\eqref{eq:cosminus}, we can obtain \begin{align} \sin x\cos y&=\frac{1}{2}\left[\sin(x+y)+\sin(x-y)\right] \label{eq:sincos}\\ \cos x\sin y&=\frac{1}{2}\left[\sin(x+y)-\sin(x-y)\right] \label{eq:cossin}\\ \cos x\cos y&=\frac{1}{2}\left[\cos(x+y)+\cos(x-y)\right] \label{eq:coscos}\\ \sin x\sin y&=\frac{1}{2}\left[\cos(x-y)-\cos(x+y)\right]. \label{eq:sinsin} \end{align}