Funciones hiperbólicas

$\begin{array}{cc}& \cosh x=\frac{e^x+e^{-x}}{2}\\ & \sinh x=\frac{e^x-e^{-x}}{2}\\ & \tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\\ & {\sin }^{2}x+{\cos }^{2}x=1={\tanh }^{2}x+\sec h^{2}x\\ & {\cosh }^{2}x=\left(\frac{e^x+e^{-x}}{2}\right)\cdot \left(\frac{e^x+e^{-x}}{2}\right)=\frac{e^{2x}+e^{-2x}+2}{4}\\ & {\sinh }^{2}x=\frac{e^{2x}+e^{-2x}-2}{4}\to {\cosh }^{2}x-{\sinh }^{2}x=1\\ \end{array}$