Arcsecant of Reciprocal equals Arccosine

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

$\blacksquare$