Challenge Level

You do not need to have met hyperbolic trig functions before in order to do this question as the definitions are given and you are led through the question step by step. You could look up the results and the proofs of the hyperbolic trig identities but you will gain much more from thinking through these proofs for yourself.

Possible extension:

Prove, by induction or otherwise, that for $x> 0$ and positive integral values of $n> 1$ $$\sinh nx > n\sinh x.$$