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