The ratio of sum of the negative and positive natural exponential functions to the subtraction of the negative natural exponential function from positive natural exponential function is called the hyperbolic cotangent function.

Take $x$ as a variable and $e$ is a positive irrational mathematical constant. The positive and negative natural exponential functions are written as $e^x$ and $e^{-x}$ respectively.

The summation of positive and negative natural exponential functions is equal to $e^x+e^{-x}$

The subtraction of negative natural exponential function from positive natural exponential function is equal to $e^x\,–\,e^{-x}$.

The ratio of summation of them to subtraction of them is written mathematically as follows.

$\large \dfrac{e^x+e^{-x}}{e^x-e^{-x}}$

The ratio is called the hyperbolic cotangent function in mathematics. The hyperbolic cotangent is represented by $\coth$ and the function is in terms of $x$. Therefore, the hyperbolic cotangent function is represented by $\coth{x}$ mathematically.

$\large \coth{x} = \dfrac{e^x+e^{-x}}{e^x-e^{-x}}$

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