Hyperbolic function

We define the hyperbolic functions for some input $x\in \mathbb{R}$:

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

Our fundamental identity is a little changed:

$${\cosh }^{2}x-{\sinh }^{2}x=1$$

And we observe a link to the fundamental trigonometric functions, for some $z\in \mathbb{C}$. All of this is provable! No need to memorise.

$$\cos z=\cosh (iz)$$ $$\cosh (z)=\cos (iz)$$ $$\sin z=\frac{\sinh (iz)}{i}$$ $$\sinh (z)=-i\sin (iz)$$