Eq. \ref{eq:4319733} is an initial equation. $$\cosh x = \frac{\exp(x) + \exp(-x)}{2} \label{eq:4319733}$$ Eq. \ref{eq:3145608} is an initial equation. $$\sinh x = \frac{\exp(x) - \exp(-x)}{2} \label{eq:3145608}$$ Multiply Eq. \ref{eq:3145608} by Eq. \ref{eq:3145608}; yields Eq. \ref{eq:7844176}. $$\sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) \label{eq:7844176}$$ Multiply Eq. \ref{eq:4319733} by Eq. \ref{eq:4319733}; yields Eq. \ref{eq:9245668}. $$\cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) \label{eq:9245668}$$ Subtract Eq. \ref{eq:7844176} from Eq. \ref{eq:9245668}; yields Eq. \ref{eq:9245668}. $$\cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) \label{eq:4001109}$$ Simplify Eq. \ref{eq:4001109}; yields Eq. \ref{eq:4009221}. $$\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right) \label{eq:4009221}$$ Simplify Eq. \ref{eq:4009221}; yields Eq. \ref{eq:6300507}. $$\cosh^2 x - \sinh^2 x = 1 \label{eq:6300507}$$ Eq. \ref{eq:6300507} is one of the final equations. Eq. \ref{eq:7404421} is an initial equation. $$\cos(i x) = \cosh(x) \label{eq:7404421}$$ Eq. \ref{eq:5377003} is an initial equation. $$\sin(i x) = i \sinh(x) \label{eq:5377003}$$ Eq. \ref{eq:3077940} is an initial equation. $$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \label{eq:3077940}$$ Change variable $$x$$ to $$i x$$ in Eq. \ref{eq:3077940}; yields Eq. \ref{eq:5823930}. $$\sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right) \label{eq:5823930}$$ LHS of Eq. \ref{eq:5823930} is equal to LHS of Eq. \ref{eq:5377003}; yields Eq. \ref{eq:2016533}. $$i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right) \label{eq:2016533}$$ Multiply both sides of Eq. \ref{eq:2016533} by $$i$$; yields Eq. \ref{eq:3145608}. $$\sinh x = \frac{\exp(x) - \exp(-x)}{2} \label{eq:3145608}$$ Eq. \ref{eq:4731536} is an initial equation. $$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) \label{eq:4731536}$$ Change variable $$x$$ to $$i x$$ in Eq. \ref{eq:4731536}; yields Eq. \ref{eq:6479977}. $$\cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right) \label{eq:6479977}$$ LHS of Eq. \ref{eq:7404421} is equal to LHS of Eq. \ref{eq:6479977}; yields Eq. \ref{eq:4319733}. $$\cosh x = \frac{\exp(x) + \exp(-x)}{2} \label{eq:4319733}$$ Eq. \ref{eq:3909583} is an initial equation. $${\rm sech}\ x = \frac{1}{\cosh x} \label{eq:3909583}$$ Substitute LHS of Eq. \ref{eq:4319733} into Eq. \ref{eq:3909583}; yields Eq. \ref{eq:7222556}. $${\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)} \label{eq:7222556}$$ Eq. \ref{eq:3867418} is an initial equation. $$\tanh(x) = \frac{\sinh(x)}{\cosh(x)} \label{eq:3867418}$$ Substitute LHS of Eq. \ref{eq:3145608} into Eq. \ref{eq:3867418}; yields Eq. \ref{eq:6831354}. $$\tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)} \label{eq:6831354}$$ Substitute LHS of Eq. \ref{eq:4319733} into Eq. \ref{eq:6831354}; yields Eq. \ref{eq:5313211}. $$\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)} \label{eq:5313211}$$ Multiply Eq. \ref{eq:5313211} by Eq. \ref{eq:5313211}; yields Eq. \ref{eq:9317216}. $$\tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} \label{eq:9317216}$$ Multiply Eq. \ref{eq:7222556} by Eq. \ref{eq:7222556}; yields Eq. \ref{eq:5395954}. $${\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} \label{eq:5395954}$$ Add Eq. \ref{eq:5395954} to Eq. \ref{eq:9317216}; yields Eq. \ref{eq:9317216}. $${\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} \label{eq:6426652}$$ Simplify Eq. \ref{eq:6426652}; yields Eq. \ref{eq:6070484}. $${\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2} \label{eq:6070484}$$ Simplify Eq. \ref{eq:6070484}; yields Eq. \ref{eq:8702257}. $${\rm sech}^2\ x + \tanh^2(x) = 1 \label{eq:8702257}$$ Eq. \ref{eq:8702257} is one of the final equations.