Inverse Trigonometric Function of Reciprocal

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

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.