Hyperbolic Secant Function is Even/Proof 2
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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 |
$\blacksquare$