Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant
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Theorem
Everywhere that the function is defined:
- $\map \arcosh {\dfrac 1 x} = \arsech x$
where $\arcosh$ and $\arsech$ denote real area hyperbolic cosine and real area hyperbolic secant respectively.
Proof
\(\ds \map \arcosh {\dfrac 1 x}\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 x\) | \(=\) | \(\ds \cosh y\) | Definition of Real Area Hyperbolic Cosine | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \sech y\) | Definition 2 of Hyperbolic Secant | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \arsech x\) | \(=\) | \(\ds y\) | Definition of Real Area Hyperbolic Secant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.62$: Relations Between Inverse Hyperbolic Functions