Arccosecant of Reciprocal equals Arcsine

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

$\blacksquare$