# Prove that (1+sinh2A+cosh2A)/(1-sinh2A-cosh2A)=-cothA?

Jun 15, 2016

See the solution below.

#### Explanation:

We will use the identities $\left(1 + \cosh 2 A\right) = 2 {\cosh}^{2} A$, $\left(1 - \cosh 2 A\right) = - 2 {\sinh}^{2} A$ and $\sinh 2 A = 2 \sinh A \cosh A$. Hence

$\frac{1 + \sinh 2 A + \cosh 2 A}{1 - \sinh 2 A - \cosh 2 A}$

= $\frac{2 {\cosh}^{2} A + 2 \sinh A \cosh A}{- 2 \sinh A \cosh A - 2 {\sinh}^{2} A}$

= $\frac{2 \cosh A \left(\cosh A + \sinh A\right)}{- 2 \sinh A \left(\cosh A + \sinh A\right)}$

= $- \frac{\cosh A}{\sinh 2 A}$

= $- \coth A$