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 \(2 x\) to \(x\) in Eq. \ref{eq:8888888}; yields Eq. \ref{eq:9999999}. \begin{equation} \exp(2 i x) = \cos(2 x)+i \sin(2 x) \label{eq:9999999} \end{equation} LHS of Eq. \ref{eq:4444444} is equal to LHS of Eq. \ref{eq:9999999}; yields Eq. \ref{eq:2222222}. \begin{equation} \cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2 \label{eq:2222222} \end{equation} Multiply Eq. \ref{eq:8888888} by Eq. \ref{eq:8888888}; yields Eq. \ref{eq:3333333}. \begin{equation} \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x)) \label{eq:3333333} \end{equation} Expand the RHS of Eq. \ref{eq:3333333}; yields Eq. \ref{eq:4444444}. \begin{equation} \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2 \label{eq:4444444} \end{equation} Select real parts of Eq. \ref{eq:2222222}; yields Eq. \ref{eq:5828294}. \begin{equation} \cos(2 x) = (\cos(x))^2 - (\sin(x))^2 \label{eq:5828294} \end{equation} Add \((\sin(x))^2\) to both sides of Eq. \ref{eq:5828294}; yields Eq. \ref{eq:4890284}. \begin{equation} \cos(2 x) + (\sin(x))^2 = (\cos(x))^2 \label{eq:4890284} \end{equation} Eq. \ref{eq:9385720} is an identity. \begin{equation} (\sin(x))^2 + (\cos(x))^2 = 1 \label{eq:9385720} \end{equation} Subtract \((\sin(x))^2\) from both sides of Eq. \ref{eq:9385720}; yields Eq. \ref{eq:9123670}. \begin{equation} (\cos(x))^2 = 1-(\sin(x))^2 \label{eq:9123670} \end{equation} Swap LHS of Eq. \ref{eq:4890284} with RHS; yields Eq. \ref{eq:2936550}. \begin{equation} (\cos(x))^2 = \cos(2 x) + (\sin(x))^2 \label{eq:2936550} \end{equation} LHS of Eq. \ref{eq:2936550} is equal to LHS of Eq. \ref{eq:9123670}; yields Eq. \ref{eq:9481000}. \begin{equation} \cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2 \label{eq:9481000} \end{equation} Subtract \((\sin(x))^2\) from both sides of Eq. \ref{eq:9481000}; yields Eq. \ref{eq:7472641}. \begin{equation} \cos(2 x) = 1 - 2 (\sin(x))^2 \label{eq:7472641} \end{equation} Add \(2(\sin(x))^2\) to both sides of Eq. \ref{eq:7472641}; yields Eq. \ref{eq:1029911}. \begin{equation} \cos(2 x) + 2 (\sin(x))^2 = 1 \label{eq:1029911} \end{equation} Subtract \(\cos(2 x)\) from both sides of Eq. \ref{eq:1029911}; yields Eq. \ref{eq:7472666}. \begin{equation} 2 (\sin(x))^2 = 1 - \cos(2 x) \label{eq:7472666} \end{equation} Divide both sides of Eq. \ref{eq:7472666} by \(2\); yields Eq. \ref{eq:1231131}. \begin{equation} (\sin(x))^2 = \frac{1 - \cos(2 x)}{2} \label{eq:1231131} \end{equation} Eq. \ref{eq:1231131} is one of the final equations.