((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