# Equivalence of Definitions of Real Area Hyperbolic Cosecant

## Theorem

The following definitions of the concept of Real Area Hyperbolic Cosecant are equivalent:

### Definition 1

The inverse hyperbolic cosecant $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:

$\forall x \in \R_{\ne 0}: \map \arcsch x := y \in \R: x = \map \csch y$

where $\map \csch y$ denotes the hyperbolic cosecant function of $y$.

### Definition 2

The inverse hyperbolic cosecant $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:

$\forall x \in \R_{\ne 0}: \map \arcsch x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$

where:

$\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$
$\ln$ denotes the natural logarithm of a (strictly positive) real number.

## Proof

### Definition 1 implies Definition 2

 $\ds x$ $=$ $\ds \csch y$ $\ds \leadsto \ \$ $\ds \dfrac 1 x$ $=$ $\ds \sinh y$ Definition of Hyperbolic Cosecant $\ds \leadsto \ \$ $\ds y$ $=$ $\ds \map \ln {\dfrac 1 x + \sqrt {\paren {\dfrac 1 x}^2 + 1} }$ Definition of Real Inverse Hyperbolic Sine $\ds$ $=$ $\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 + x^2} {x^2} } }$

If $x \ge 0$, then:

 $\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 + x^2} {x^2} } }$ $=$ $\ds \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } x}$ Positive Square Root is Positive

If $x < 0$, then:

 $\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 + x^2} {x^2} } }$ $=$ $\ds \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {-x} }$ Positive Square Root is Positive

That is:

$y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }$

$\Box$

### Definition 2 implies Definition 1

 $\ds y$ $=$ $\ds \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }$ $\ds$ $=$ $\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac {1 + x^2} {x^2} } }$ $\ds$ $=$ $\ds \map \ln {\dfrac 1 x + \sqrt {\dfrac 1 {x^2} + 1} }$ $\ds$ $=$ $\ds \arsinh \dfrac 1 x$ Definition of Real Area Hyperbolic Sine $\ds \leadsto \ \$ $\ds \dfrac 1 x$ $=$ $\ds \sinh y$ Definition of Hyperbolic Sine $\ds \leadsto \ \$ $\ds x$ $=$ $\ds \csch y$ Definition of Hyperbolic Cosecant

$\Box$

Therefore:

 $\text {(1)}: \quad$ $\ds x = \csch y$ $\implies$ $\ds y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }$ Definition 1 implies Definition 2 $\text {(2)}: \quad$ $\ds y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }$ $\implies$ $\ds x = \csch y$ Definition 2 implies Definition 1 $\ds \leadsto \ \$ $\ds x = \csch y$ $\iff$ $\ds y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }$

$\blacksquare$