Equivalence of Definitions of Hyperbolic Cosecant

Theorem

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

Definition 1

The hyperbolic cosecant function is defined on the complex numbers as:

$\csch: X \to \C$:
$\forall z \in X: \csch z := \dfrac 2 {e^z - e^{-z} }$

where:

$X = \set {z: z \in \C, \ e^z - e^{-z} \ne 0}$

Definition 2

The hyperbolic cosecant function is defined on the complex numbers as:

$\csch: X \to \C$:
$\forall z \in X: \csch z := \dfrac 1 {\sinh z}$

where:

$\sinh$ is the hyperbolic sine
$X = \set {z: z \in \C, \ \sinh z \ne 0}$

Proof

 $\displaystyle \forall z \in \left\{ {z \in \C: \ e^z - e^{-z} \ne 0}\right\}: \ \$ $\displaystyle$  $\displaystyle \frac 2 {e^z - e^{-z} }$ Definition of Hyperbolic Cosecant: Definition 1 $\displaystyle \forall z \in \left\{ {z \in \C: \ \frac {e^z - e^{-z} } 2 \ne 0}\right\}: \ \$ $\displaystyle$ $=$ $\displaystyle 1 / \frac {e^z - e^{-z} } 2$ $\displaystyle \forall z \in \left\{ {z \in \C: \ \sinh z \ne 0}\right\}: \ \$ $\displaystyle$ $=$ $\displaystyle 1 / \sinh z$ Definition of Hyperbolic Cosecant: Definition 2

$\blacksquare$