Answer
$\sqrt2.$
Work Step by Step
Note that $\tan^{-1}{1}=\dfrac{\pi}{4}$, because $\tan{\left(\dfrac{\pi}{4}\right)}=1$ and $\dfrac{\pi}{4}$ is in the range of $\tan^{-1}{x}$, which is $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right].$
Thus
$\csc{\left(\tan^{-1}{\left(1\right)}\right)}=\csc{\left(\dfrac{\pi}{4}\right)}=\sqrt2.$