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.

Notice that the identities for hyperbolic functions that you have proved are very similar to the ordinary trigonometric identities. In fact there is a complete hyperbolic geometry with similar results to the trigonometric results in Euclidean geometry. We compare absolute values in the corresponding result for $\sin \mathrm{nx}$ which is $|\sin \mathrm{nx}|\le n|\sin x|$ . This formula needs the absolute values because the function is periodic and takes negative values for some multiples of the angle. Notice that the inequality in $|\sin \mathrm{nx}|\le n|\sin x|$ goes the other way to the corresponding hyperbolic result. This is because $\cos x\le 1$ for all $x$ whereas $\cosh x\ge 1$.