Generated by the Physics Derivation Graph. Eq. \ref{eq:8888888} is an initial equation. \begin{equation} \exp(i x) = \cos(x)+i \sin(x) \label{eq:8888888} \end{equation} Change variable \(x\) to \(-x\) in Eq. \ref{eq:8888888}; yields Eq. \ref{eq:8888881}. \begin{equation} \exp(-i x) = \cos(-x)+i \sin(-x) \label{eq:8888881} \end{equation} $x\) is even with respect to \(\cos(x)$, so replace \(x\) with \(\cos(-x)\) in Eq. \ref{eq:8888881}; yields Eq. \ref{eq:8888882}. \begin{equation} \exp(-i x) = \cos(x)+i \sin(-x) \label{eq:8888882} \end{equation} $x\) is odd with respect to \(-\sin(x)$, so replace \(x\) with \(\sin(-x)\) in Eq. \ref{eq:8888882}; yields Eq. \ref{eq:8888883}. \begin{equation} \exp(-i x) = \cos(x)-i \sin(x) \label{eq:8888883} \end{equation} Add Eq. \ref{eq:8888888} to Eq. \ref{eq:8888883}; yields Eq. \ref{eq:8888883}. \begin{equation} \exp(i x)+\exp(-i x) = 2 \cos(x) \label{eq:2939484} \end{equation} Divide both sides of Eq. \ref{eq:2939484} by \(2\); yields Eq. \ref{eq:4383592}. \begin{equation} \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x) \label{eq:4383592} \end{equation} Swap LHS of Eq. \ref{eq:4383592} with RHS; yields Eq. \ref{eq:4849888}. \begin{equation} \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) \label{eq:4849888} \end{equation} Eq. \ref{eq:4849888} is one of the final equations. Multiply both sides of Eq. \ref{eq:8888883} by \(-1\); yields Eq. \ref{eq:3194924}. \begin{equation} -\exp(-i x) = -\cos(x)+i \sin(x) \label{eq:3194924} \end{equation} Add Eq. \ref{eq:8888888} to Eq. \ref{eq:3194924}; yields Eq. \ref{eq:3194924}. \begin{equation} \exp(i x)-\exp(-i x) = 2 i \sin(x) \label{eq:4825483} \end{equation} Divide both sides of Eq. \ref{eq:4825483} by \(2 i\); yields Eq. \ref{eq:1133483}. \begin{equation} \frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x) \label{eq:1133483} \end{equation} Swap LHS of Eq. \ref{eq:1133483} with RHS; yields Eq. \ref{eq:4849959}. \begin{equation} \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \label{eq:4849959} \end{equation}