Hyperbolic trig functions

The Hyperbolic trig functions can be thought of the classical trig functions except found on an unit hyperbola. There are as follows:

$\sinh(x)=\frac{e^x+e^{-x}}{2}$

$\cosh(x)=\frac{e^x-e^{-x}}{2}$

$\tanh(x)= \frac{\sinh{x}}{\cosh{x}} =\frac{e^x+e^{-x}}{e^x-e^{-x}}$

They behave much like normal trig functions, as most of the identities still hold. They do so because:

$\sinh(x)= -i\sin{ix}$

$\cosh(x)=\cos{iz}$

$\tanh(x)= -1\tan{iz}$

However, the arc-hyperbolic trig functions have different derivatives.

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