Answer
$${\coth ^2}x - 1 = {\operatorname{csch} ^2}x$$
Work Step by Step
$$\eqalign{
& {\cosh ^2}x - {\sinh ^2}x = 1 \cr
& {\text{divide by sin}}{{\text{h}}^2}x \cr
& \frac{{{{\cosh }^2}x}}{{{\text{sin}}{{\text{h}}^2}x}} - \frac{{{{\sinh }^2}x}}{{{\text{sin}}{{\text{h}}^2}x}} = \frac{1}{{{\text{sin}}{{\text{h}}^2}x}} \cr
& {\text{simplify}} \cr
& \frac{{{{\cosh }^2}x}}{{{\text{sin}}{{\text{h}}^2}x}} - 1 = \frac{1}{{{\text{sin}}{{\text{h}}^2}x}} \cr
& {\text{use hyperbolic functions cosecant and cotangent}} \cr
& {\coth ^2}x - 1 = {\operatorname{csch} ^2}x \cr} $$
