Hyperbolic identities

Hyperbolic Functions Introduction

Hyperbolic functions, often coined as hyperbolics, extend the notion of the trigonometric functions to the complex plane.
Hyperbolic functions are analogues to the ordinary trigonometric, or circular, functions.
The basic hyperbolic functions are: hyperbolic sine (sinh), hyperbolic cosine (cosh), hyperbolic tangent (tanh), hyperbolic cotangent (coth), hyperbolic secant (sech), and hyperbolic cosecant (csch).

Sinh(x) is defined as (e^x - e^-x) / 2, it describes the shape of a hanging cable or chain, the so-called catenary.
Cosh(x) is defined as (e^x + e^-x) / 2, it describes the shape of a suspension bridge.
Other hyperbolic functions like tanh(x), coth(x), sech(x), csch(x) are defined as the ratios of sinh(x) and cosh(x) similar to how we define other trigonometric functions.

Similar to the Pythagorean identity in trigonometry, there is an identity for hyperbolic functions: cosh²(x) - sinh²(x) = 1.
The sum of sinh is given by sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y).
The sum of cosh is given by cosh(x + y) = cosh(x)cosh(y) + sinh(x)sinh(y).

Graphs of hyperbolic functions are called hyperbolas.
The graph of sinh(x) is a smooth, increasing curve; while the graph of cosh(x) looks like a “U” shape with its vertex at (0,1).
Just as regular trigonometric functions are related to the unit circle, hyperbolic functions are related to a hyperbola.

Hyperbolic functions don’t have a finite period like trigonometric functions, they are unbounded and increase or decrease monotonically.
Hyperbolic functions have similar identities, related graphs and properties that correspond to trigonometric functions but they differ due to the distinguishing minus sign in their identities.
These functions are central to understanding certain characteristics of nature and appear in various areas of maths including complex numbers, calculus and differential equations.