Hyperbolic Functions

Hyperbolic cosine is color(red)(y=cosh(x)=(e^x+e^(-x))/2).

Hyperbolic sine is color(blue)(y=sinh(x)=(e^x-e^(-x))/2).

Hyperbolic tangent is color(green)(y=tanh(x)=(sinh(x))/(cosh(x))=(e^x-e^(-x))/(e^x+e^(-x))).

Hyperbolic cotangent is y=coth(x)=(cosh(x))/(sinh(x)=(e^x+e^(-x))/(e^x-e^(-x))).

Hyperbolic secant is y=text(sech)(x)=1/(cosh(x))=2/(e^x+e^(-x)) .

Hyperbolic cosecant is y=csch(x)=1/(sinh(x))=2/(e^x-e^(-x)).

There is some similarity between hyperbolic functions and trigonometric.

Domain of hyperbolic functions is (-oo,oo), except for function y=coth(x) which is undefined when x=0.

Formulas that hold for any x and y:

1. cosh(x+-y)=cosh(x)cosh(y)+-sinh(x)sinh(y).
2. sinh(x+-y)=sinh(x)cosh(y)+-cosh(x)sinh(y).
3. cosh^2(x)-sinh^2(x)=1.
4. cosh(2x)=cosh^2(x)+sinh^2(x).
5. sinh(2x)=2sinh(x)cosh(x).

This formulas can be easily proved using definitions of hyperbolic functions.