Equivalence of Definitions of Real Hyperbolic Cotangent

Theorem

The following definitions of the concept of Real Hyperbolic Cotangent are equivalent:

Definition 1

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

$\coth: \R_{\ne 0} \to \R$:

$\forall x \in \R_{\ne 0}: \coth x := \dfrac {e^x + e^{-x} } {e^x - e^{-x} }$

where it is noted that at $x = 0$:

$e^x - e^{-x} = 0$

and so $\coth x$ is not defined at that point.

Definition 2

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.

Definition 3

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

$\coth: \R_{\ne 0} \to \R$:

$\forall x \in \R_{\ne 0}: \coth x := \dfrac 1 {\tanh x}$

where $\tanh$ is the real hyperbolic tangent.

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

Proof

\(\ds \coth x\)

\(=\)

\(\ds \dfrac {\cosh x} {\sinh x}\)

Definition 2 of Real Hyperbolic Cotangent

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {\dfrac {e^x - e^{-x} } 2} } {\paren {\dfrac {e^x + e^{-x} } 2} }\)

Definition of Real Hyperbolic Sine, Definition of Real Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \dfrac {e^x + e^{-x} } {e^x - e^{-x} }\)

simplification

\(\ds \)

\(=\)

\(\ds \coth x\)

Definition 1 of Real Hyperbolic Cotangent

$\Box$

\(\ds \coth x\)

\(=\)

\(\ds \dfrac {\cosh x} {\sinh x}\)

Definition 2 of Real Hyperbolic Cotangent

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\paren {\dfrac {\sinh x} {\cosh x} } }\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\tanh x}\)

Definition 2 of Real Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \coth x\)

Definition 3 of Real Hyperbolic Cotangent

$\blacksquare$