Hyperbolic functions are similar to sin,cos tan etc in trigonometry and share many similar rules. Usually hyperbolic functions are written like the trigonometric ones but with a h on the end, eg sinh and cosh.

The hyperbolic functions can be all written in terms of e, sinh and cosh are as follows

sinh(x) = (e^x – e^-x)/2
cosh(x) = (e^x + e^-x)/2

And tanh can be defined as sinh/cosh so

tanh = (e^x – e^-x) / (e^x + e^-x)

though this is often written as
tanh = (e^2x – 1) / (e^2x + 1)
by timesing the top and bottom by e^x

the other other hyperbolic functions sinh as sech, coth etc can be found in the same way as they would be in trigonometry, by using 1 over the other functions, ie sech = 1/cosh

Most of the identities in trigonometry have a similar identity with hyperbolic functions, however in most of these whenever there is a sin² it changes to a -sinh²
so
cosh² – sinh²=1
which you can work out by placing the equations with e’s in the place of sinh and cosh