Inverse Trigonometric Function of Reciprocral

Theorem

$\arcsin x = \operatorname{arccsc}\dfrac 1 x$

$\operatorname{arccsc} x = \arcsin \dfrac 1 x$

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

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

$\arctan x = \operatorname{arccot} \dfrac 1 x$

$\operatorname{arccot} x = \arctan \dfrac 1 x$

for all $x \in \R$ for which the expressions above are defined.

Proof

Let $y \in \left[{-\dfrac \pi 2 .. \dfrac \pi 2}\right] \setminus \{0\}$.

By definition of arcsine:

$x = \sin y \iff \arcsin x = y$

By definition of cosecant and arccosecant:

$\dfrac 1 x = \csc y \iff \operatorname{arccsc} \dfrac 1 x = y$

$\implies \arcsin x = \operatorname{arccsc}\dfrac 1 x$

The proofs of the other identities are similar.

$\blacksquare$