Answer
$coth^2~x-1 = csch^2~x$
Work Step by Step
$coth^2~x-1 = (\frac{cosh~x}{sinh~x})^2-1$
$coth^2~x-1 = [\frac{(e^x+e^{-x})/2}{(e^x-e^{-x})/2}]^2-1$
$coth^2~x-1 = [\frac{e^x+e^{-x}}{e^x-e^{-x}}]^2-1$
$coth^2~x-1 = \frac{e^{2x}+2+e^{-2x}}{e^{2x}-2-e^{-2x}}-1$
$coth^2~x-1 = \frac{e^{2x}+2+e^{-2x}}{e^{2x}-2-e^{-2x}}- \frac{e^{2x}-2+e^{-2x}}{e^{2x}-2-e^{-2x}}$
$coth^2~x-1 = \frac{4}{e^{2x}-2-e^{-2x}}$
$coth^2~x-1 = (\frac{2}{e^x-e^{-x}})^2$
$coth^2~x-1 = (\frac{1}{sinh~x})^2$
$coth^2~x-1 = csch^2~x$
