Hyperbolic trigonometric functions

The hyperbolic trigonometric functions are analogs of the standard trigonometric functions using a hyperbola as the defining conic section rather than a circle. They are defined as:

• Hyperbolic sine: $sinh(x)={\frac {e^{x}-e^{-x}}{2}}$
• Hyperbolic cosine: $cosh(x)={\frac {e^{x}+e^{-x}}{2}}$
• Hyperbolic tangent: $tanh(x)={\frac {sinh(x)}{cosh(x)}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}$
• Hyperbolic cosecant: $csch(x)={\frac {1}{sinh(x)}}={\frac {2}{e^{x}-e^{-x}}}$
• Hyperbolic secant: $sech(x)={\frac {1}{sinh(x)}}={\frac {2}{e^{x}+e^{-x}}}$
• Hyperbolic tangent: $coth(x)={\frac {cosh(x)}{sinh(x)}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}$