The hyperbolic trig functions $\cosh $ and $\sinh $ are defined by
$$\eqalign { \cosh x &= {1\over 2}(e^x + e^{-x}) \cr \sinh x
&= {1\over 2}(e^x - e^{-x}).}$$ Using the definitions sketch
the graphs of $\cosh x$ and $\sinh x$ on one diagram and prove the
hyperbolic trig identities $$\eqalign { \cosh^2 x - \sinh^2 x
&=1 \cr \sinh 2x &= 2\sinh x \cosh x \cr \sinh (n+1)x
&= \sinh nx \cosh x + \cosh nx \sinh x.}$$
Notice the strong resemblance of these formulae to standard
trigonometrical identities. Using this similarity as a guide,
investigate the properties of a 'hyperbolic tangent' function
$tanh(x)$ defined by
$$\tanh(x)=\frac{\sinh(x)}{\cosh(x)}$$
NOTES AND BACKGROUND
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 $\sin nx$
which is $|\sin nx|\leq n|\sin x|$ . 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 $|\sin nx|\leq n|\sin x|$ goes the other way to the
corresponding hyperbolic result. This is because $\cos x \leq 1$
for all $x$ whereas $\cosh x\geq 1$.