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