How did mathematicians discover the expressions of hyperbolic functions?

[Leo Liu]

The hyperbolic function $\cosh t \text{ and } \sinh t$ respectively represent the x and y coordinate of the parametric equation of the parabola $x^2-y^2=1$. The exponential expressions of these hyperbolic functions are
$$
\begin{cases}
\sinh x = {e^x-e^{-x}} /2; \: x \in \mathbb R, \: f(x) \in \mathbb R \\
\cosh x = {e^x+e^{-x}} /2; \: x \in \mathbb R, \: f(x) \in [1,\infty)
\end{cases}
$$
But I would like to know how to derive these expressions. Thanks.

P.S. I do know the proof--$\cosh^2 t - \sinh^2 t = 1$ is equivalent to $x^2-y^2=1$.

 

[hutchphd]

This would be easier to follow if you always call the parameter t and the Cartesian coordinates (x,y)