User:Rushikeshjogdand1

Hyperbolic Functions

Every function $f(x)$ can be written in a unique way as composition of an even function and an odd functions — $f(x)={\frac {f(x)+f(-x)}{2}}+{\frac {f(x)-f(-x)}{2}}$.

Similarly the exponential function $e^{x}$ can be displayed as — $e^{x}={\frac {e^{x}+e^{-x}}{2}}+{\frac {e^{x}-e^{-x}}{2}}$. The even and odd parts are called as hyperbolic cosine and hyperbolic sine of $x$.

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

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