Arccosecant of Reciprocal equals Arcsine
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Theorem
Everywhere that the function is defined:
- $\map \arccsc {\dfrac 1 x} = \arcsin x$
where $\arcsin$ and $\arccsc$ denote arcsine and arccosecant respectively.
Proof
\(\ds \map \arccsc {\frac 1 x}\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 x\) | \(=\) | \(\ds \csc y\) | Definition of Real Arccosecant | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \sin y\) | Cosecant is Reciprocal of Sine | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \arcsin x\) | \(=\) | \(\ds y\) | Definition of Real Arcsine |
$\blacksquare$