Equivalence of Definitions of Hyperbolic Secant

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Theorem

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

Definition 1

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

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

where:

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

Definition 2

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

$\sech: X \to \C$:
$\forall z \in X: \sech z := \dfrac 1 {\cosh z}$

where:

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


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 Secant: Definition 1
\(\displaystyle \forall z \in \left\{ {z \in \C: \ \frac {e^z + e^{-z} } 2 \ne 0}\right\}: \ \ \) \(\displaystyle \) \(=\) \(\displaystyle 1 / \frac {e^z + e^{-z} } 2\)
\(\displaystyle \forall z \in \left\{ {z \in \C: \ \cosh z \ne 0}\right\}: \ \ \) \(\displaystyle \) \(=\) \(\displaystyle 1 / \cosh z\) Definition of Hyperbolic Secant: Definition 2

$\blacksquare$