Answer
$\theta=\{120^o,240^o\}$
Work Step by Step
$csc(\theta)+2cot(\theta)=0$
$\frac{1}{sin(\theta)}+2\frac{cos(\theta)}{sin(\theta)}=0\;\;\;\;\;\;\;\;\;\;$ multiply each side by $sin(\theta)$.
$1+2cos(\theta)=0$
$2cos(\Theta )=-1\;\;\;\;\;\;\;\;\;\;$ subtract $ 1 $ from each side.
$cos(\Theta )=\frac{-1}{2} \;\;\;\;\;\;\;\;\;\;\;$ divide each side by $ 2 $
$cos(\Theta )=\frac{-1}{2}$
$\theta= cos^{-1}(\frac{-1}{2})$
We know $ cos(\theta) $ is negative in quadrant $II$ and quadrant $III$
$\theta=180^o-60^o=120^o\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\theta=180^o+60^o=240^o$
$\theta=120^o\;\;\;\;\;\;\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\;\;\;\theta=240^o$
$\theta=\{120^o,240^o\}$
