# Definition:Real Hyperbolic Cotangent/Definition 2

## Definition

The real hyperbolic cotangent function is defined on the real numbers as:

$\coth: \R_{\ne 0} \to \R$:
$\forall x \in \R_{\ne 0}: \coth x := \dfrac {\cosh x} {\sinh x}$

where:

$\sinh$ is the real hyperbolic sine
$\cosh$ is the real hyperbolic cosine

It is noted that at $x = 0$ we have that $\sinh x = 0$, and so $\coth x$ is not defined at that point.