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
- Hyperbolic Sine Function is Odd
- Hyperbolic Cosine Function is Even
- Hyperbolic Tangent Function is Odd
- Hyperbolic Cotangent Function is Odd
- Hyperbolic Cosecant Function is Odd
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.18$: Functions of Negative Arguments