Every function \(f(x)\) can be written uniquely as sum of an odd part and an even part.

\[f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x) - f(-x)}{2}\]
\[\therefore e^{x}=\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}\]
The even and odd terms are called as *hyperbolic cosine* and *hyperbolic sine* function of \(x\).

i.e.

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

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