hyperbolic sine:	sinh x = (1/2) (ex - e-x)
hyperbolic cosine:	cosh x = (1/2) (ex + e-x)
hyperbolic tangent:	tanh x = sinh x / cosh x
hyperbolic cotangent:	coth x = 1 / tanh x
hyperbolic secant:	sech x = 1 / cosh x
hyperbolic cosecant:	cosech x = 1 / sinh x

They are called hyperbolic functions because in some ways they have properties similar to the trigonometric functions and they are related to the hyperbola in the way that the circular functions (trigonometric functions) are related to the circle.The following is a list of some of the fundamental relationships between hyperbolic functions:

sinh (-x) = -sinh (x)
cosh (-x) = +cosh (x)
cosh²x - sinh²x = 1
sech²x + tanh²x = 1
coth²x - cosech²x = 1