Answer
$a.\displaystyle \quad\frac{\pi}{4}$
$b.\displaystyle \quad\frac{\pi}{6}$
Work Step by Step
$ a.\quad$
$y=\csc^{-1}x,\ (|x|\geq 1) \ \Leftrightarrow\ \csc y=x\ $ and $\ y\in(0, \pi/2] \cup(\pi, 3\pi/2]$
$y=\displaystyle \frac{\pi}{4}\in(0, \pi/2]$ is such that $\displaystyle \csc\frac{\pi}{4}=\sqrt{2}$, so $\csc^{-1}\sqrt{2}$=$\displaystyle \frac{\pi}{4}$
$ b.\quad$
$\cos^{-1}x=y \ \Leftrightarrow\ \cos y=x\ $ and $\ y\in[0,\pi]$
$y=\displaystyle \frac{\pi}{6}\in[0,\pi]$ is such that $\displaystyle \cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$, so $\displaystyle \cos^{-1}(\frac{\sqrt{3}}{2})$=$\displaystyle \frac{\pi}{6}$
