# Hyperbolic cosecant function

The ratio of number $2$ to subtraction of the negative natural exponential function from the positive natural exponential function is called the hyperbolic cosecant function.

## Introduction

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

The subtraction of the negative natural exponential function from positive natural exponential function is $e^x-e^{-x}$

The ratio of the quantity $2$ to the subtraction of them is written mathematically as follows.

$\large \dfrac{2}{e^x-e^{-x}}$

In mathematics, the ratio of $2$ to the subtraction of the natural exponential functions is called the hyperbolic co-secant function. It is denoted by either ${\mathop{\rm cosech}\nolimits}$ or ${\mathop{\rm csch}\nolimits}$ but the function is expressed in terms of $x$. Therefore, the hyperbolic cosecant function is written as ${\mathop{\rm cosech}\nolimits}{x}$ or ${\mathop{\rm csch}\nolimits}{x}$ mathematically.

$\large {\mathop{\rm csch}\nolimits}{x} \,=\, \dfrac{2}{e^x-e^{-x}}$

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