Use Osborn's Rule to suggest an identity involving cothx^2 and cosechx^2 then to solve

Question

i know 1+cothx^2=cschx^2

but how to suggest the identity then use to solve part b?

Answer 1

If you replace $\text{cosech}^2 x$ by $1 + \coth^2 x$, you get a quadratic equation in $\coth x$.