((e^x-e^(-x))/2)^2-1

((e^x-e^(-x))/2)^2-1

$$\left( \dfrac{e^x-e^{-x}}{2} \right) ^2 -1 =\sinh^2{x}-1 = \cosh^2{x}-2 = \left( \dfrac{e^x+e^{-x}}{2} \right) ^2 -2 \\\\ ( \cosh^2{x}-\sinh^2{x} =1 )$$

$$\left( \frac{e^x-e^{-x}}{2}\right)^2-1\\\\ = \frac{(e^x-e^{-x})^2}{2^2}-1\\\\ = \frac{e^{2x}-2e^xe^{-x}+e^{-2x}}{4}-\frac{4}{4}\\\\ = \frac{e^{2x}-2+e^{-2x}}{4}-\frac{4}{4}\\\\ = \frac{e^{2x}+e^{-2x}-6}{4}\\\\ \mbox{Do you want me to keep going? It is probably finish now:)}\\\\ = \frac{e^{2x}+\frac{1}{e^{2x}}-6}{4}\\\\ = \frac{\frac{e^{4x}+1-6e^{2x}}{e^{2x}}}{4}\\\\ = \frac{e^{4x}+1-6e^{2x}}{4e^{2x}}\\\\ = \frac{e^{4x}-6e^{2x}+1}{4e^{2x}}\\\\$$

Thanks Heureka,

I had forgotten about cosh and sinh