Generated by the Physics Derivation Graph. Eq. \ref{eq:6060683} is an initial equation. $$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \label{eq:6060683}$$ Change variable $$x$$ to $$2 x$$ in Eq. \ref{eq:6060683}; yields Eq. \ref{eq:1414263}. $$\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right) \label{eq:1414263}$$ Eq. \ref{eq:5011637} is an initial equation. $$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) \label{eq:5011637}$$ Multiply Eq. \ref{eq:6060683} by Eq. \ref{eq:5011637}; yields Eq. \ref{eq:6350246}. $$\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) \label{eq:6350246}$$ Multiply both sides of Eq. \ref{eq:6350246} by $$2$$; yields Eq. \ref{eq:7867574}. $$2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right) \label{eq:7867574}$$ Simplify Eq. \ref{eq:7867574}; yields Eq. \ref{eq:5714636}. $$2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right) \label{eq:5714636}$$ Simplify Eq. \ref{eq:5714636}; yields Eq. \ref{eq:6229292}. $$2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right) \label{eq:6229292}$$ RHS of Eq. \ref{eq:6229292} is equal to RHS of Eq. \ref{eq:1414263}; yields Eq. \ref{eq:7647794}. $$\sin(2 x) = 2 \sin(x) \cos(x) \label{eq:7647794}$$ Eq. \ref{eq:7647794} is one of the final equations.