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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##.

$$

\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##.

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