Should be relatively simple but not sure what to do after a few attempts...
$\displaystyle q(x)=csc^{ - 1}(x^{ - 1})$
not much to see, other than knowing that sine and cosecant are reciprocals ...
if $\displaystyle x = \sin{\theta}$ , then $\displaystyle \frac{1}{x} = \csc{\theta}$
inverses ...
$\displaystyle \sin^{-1}(x) = \theta$ ... $\displaystyle \csc^{-1}\left(\frac{1}{x}\right) = \theta$