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