Arccosine of Reciprocal equals Arcsecant
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Theorem
Everywhere that the function is defined:
- $\map \arccos {\dfrac 1 x} = \arcsec x$
where $\arccos$ and $\arcsec$ denote arccosine and arcsecant respectively.
Proof
| \(\ds \map \arccos {\frac 1 x}\) | \(=\) | \(\ds y\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 x\) | \(=\) | \(\ds \cos y\) | Definition of Real Arccosine | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \sec y\) | Secant is Reciprocal of Cosine | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \arcsec x\) | \(=\) | \(\ds y\) | Definition of Real Arcsecant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.78$