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