Arcsecant of Reciprocal equals Arccosine
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Theorem
Everywhere that the function is defined:
- $\map \arcsec {\dfrac 1 x} = \arccos x$
where $\arccos$ and $\arcsec$ denote arccosine and arcsecant respectively.
Proof
\(\ds \map \arcsec {\frac 1 x}\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 x\) | \(=\) | \(\ds \sec y\) | Definition of Real Arcsecant | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \cos y\) | Secant is Reciprocal of Cosine | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \arccos x\) | \(=\) | \(\ds y\) | Definition of Real Arcsecant |
$\blacksquare$