Answer
$\frac{\pi}{4}$, $\frac{3\pi}{4}$, $\frac{7\pi}{6}$, $\frac{11\pi}{6}$
Work Step by Step
$(\csc x+2)(\csc x-\sqrt{2})=0$
$\csc x+2=0$ or $\csc x-\sqrt{2}=0$
If $\csc x+2=0$:
$\csc x=-2$
$\frac{1}{\sin x}=-2$
$\sin x=-\frac{1}{2}$
The only solutions in $[0, 2\pi)$ are $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$.
If $\csc x-\sqrt{2}=0$:
$\csc x=\sqrt{2}$
$\frac{1}{\sin x}=\sqrt{2}$
$\sin x=\frac{1}{\sqrt{2}}$
$\sin x=\frac{\sqrt{2}}{2}$
The only solutions in $[0, 2\pi)$ are $\frac{\pi}{4}$ and $\frac{3\pi}{4}$.
