Hyperbolic cosine and hyperbolic sine, denoted by cosh(x) and sinh(x) are, respectively, the even

and odd terms in the series expansion for exp(x).

(The ordinary trigonometric functions are evenand (odd part)/i of exp(ix).) 

Other hyperbolic functions such as tanh(x), coth(x),and the inverse of all of these are defined

from cosh(x) and sinh(x) exactly as tan(x), cot(x)and thecorresponding inverses are from cosineand sine.This implies the following: 

Relation to the exponent:

Series expansions:

Pythagorian analogue:

cosh²x = sinh²x + 1

Differential formulae:

There are addition theorems and half angle formulae exactly analoguous to those for ordinary trigonometric functions.