The hyperbolic trig functions $\cosh$ and $\sinh$ are defined by

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{\cosh x}& =\frac{1}{2}(e^x+e^{-x})\\ \multicolumn{1}{c}{\sinh x}& =\frac{1}{2}(e^x-e^{-x}).\end{array}}$

Using the definitions sketch the graphs of $\cosh x$ and $\sinh x$ on one diagram and prove the hyperbolic trig identities

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{{\cosh }^{2}x-{\sinh }^{2}x}& =1\\ \multicolumn{1}{c}{\sinh 2x}& =2\sinh x\cosh x\\ \multicolumn{1}{c}{\sinh (n+1)x}& =\sinh \mathrm{nx}\cosh x+\cosh \mathrm{nx}\sinh x.\end{array}}$

Prove, by induction or otherwise, that for $x>0$ and positive integral values of $n$

${\displaystyle \sinh \mathrm{nx}\geqq n\sinh x.}$