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