Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant
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Theorem
Everywhere that the function is defined:
- $\map \arsinh {\dfrac 1 x} = \arcsch x$
where $\arsinh$ and $\arcsch$ denote real area hyperbolic sine and real area hyperbolic cosecant respectively.
Proof
| \(\ds \map \arsinh {\dfrac 1 x}\) | \(=\) | \(\ds y\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 x\) | \(=\) | \(\ds \sinh y\) | Definition of Real Area Hyperbolic Sine | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \csch y\) | Definition 2 of Hyperbolic Cosecant | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \arcsch x\) | \(=\) | \(\ds y\) | Definition of Real Area Hyperbolic Cosecant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.61$: Relations Between Inverse Hyperbolic Functions