# Hyperbolic Functions

Let's see the basics of the hyperbolic functions (sinh x, coshx): definition, property, and the proof of the property.

## Definition

Hyperbolic sine:

sinh *x* is defined as (*e*^{x} - *e*^{-x}) / 2.

Hyperbolic cosine:

cosh *x* is defined as (*e*^{x} + *e*^{-x}) / 2.

## Property

Recall that

the point on a unit circle, *x*^{2} + *y*^{2} = 1,

is (cos *x*, sin *x*).

Point on a unit circle (using sine and cosine)

Similiarly,

the point on a unit hyperbola, *x*^{2} - *y*^{2} = 1,

is (cosh *x*, sinh *x*).

This is why the letter 'h' is added.

So cosh^{2} *x* - sinh^{2} *x* = 1.

## Proof

cosh *x* = (*e*^{x} + *e*^{-x}) / 2

sinh *x* = (*e*^{x} - *e*^{-x}) / 2

Then cosh^{2} *x* - sinh^{2} *x* = [(*e*^{x} + *e*^{-x}) / 2]^{2} - [(*e*^{x} - *e*^{-x}) / 2]^{2}.

Square of a sum

Square of a difference

Cancel the dark gray terms.

So cosh^{2} *x* - sinh^{2} *x* = 1.