Arccosine of Reciprocal equals Arcsecant

Theorem

Everywhere that the function is defined:

$\map \arccos {\dfrac 1 x} = \arcsec x$

where $\arccos$ and $\arcsec$ denote arccosine and arcsecant respectively.

$\ds \map \arccos {\frac 1 x}$

$=$

$\ds y$

$\ds \leadstoandfrom \ \ $

$\ds \frac 1 x$

$=$

$\ds \cos y$

Definition of Real Arccosine

$\ds \leadstoandfrom \ \ $

$\ds x$

$=$

$\ds \sec y$

$\ds \leadstoandfrom \ \ $

$\ds \arcsec x$

$=$

$\ds y$

Definition of Real Arcsecant

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.78$