Hyperbolic Secant Function is Even/Proof 2

Theorem

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


Proof

\(\ds \sech \paren {-x}\) \(=\) \(\ds \frac 1 {\cosh \paren {-x} }\) 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

$\blacksquare$

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