Arcsine of Reciprocal equals Arccosecant

Theorem
Let $x \in \R: \left|{x}\right| \ge 1$.

Then:
 * $\arcsin \left({\dfrac 1 x}\right) = \operatorname{arccsc} x$

where $\arcsin$ and $\operatorname{arccsc}$ denote arcsine and arccosecant respectively.

Proof
By definition, arccosecant is defined on the real numbers only when $\left|{x}\right| \ge 1$.

Then: