Hyperbolic Secant Function is Even

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Theorem

Let $\sech: \C \to \C$ be the hyperbolic secant function on the set of complex numbers.


Then $\sech$ is even:

$\map \sech {-x} = \sech x$


Proof 1

\(\ds \map \sech {-x}\) \(=\) \(\ds \frac 1 {\map \cosh {-x} }\) Definition 2 of Hyperbolic Secant
\(\ds \) \(=\) \(\ds \frac 1 {\cosh x}\) Hyperbolic Cosine Function is Even
\(\ds \) \(=\) \(\ds \sech x\)

$\blacksquare$


Proof 2

\(\ds \sech \paren {-x}\) \(=\) \(\ds \frac 1 {\cosh \paren {-x} }\) Definition of Hyperbolic Secant
\(\ds \) \(=\) \(\ds \frac 1 {\cos \paren {-i x} }\) Hyperbolic Cosine in terms of Cosine
\(\ds \) \(=\) \(\ds \frac 1 {\cos \paren {i x} }\) Cosine Function is Even
\(\ds \) \(=\) \(\ds \frac 1 {\cosh x}\) Hyperbolic Cosine in terms of Cosine
\(\ds \) \(=\) \(\ds \sech x\) Definition of Hyperbolic Secant

$\blacksquare$


Also see


Sources