Answer
$\theta=\{45^o,135^o,225^o,315^o\}$
Work Step by Step
$4sin(\theta)-2csc(\theta)=0$
$4sin(\theta)-2\frac{1}{sin(\theta)}=0\;\;\;\;\;\;\;\;\;\;$ multipl each side by $sin(\theta)$.
$4sin^2(\theta)-2=0$
$4sin^2(\Theta )=2\;\;\;\;\;\;\;\;\;\;$ subtract $ -2 $ from each side.
$sin^2(\Theta )=\frac{2}{4}=\frac{1}{2} \;\;\;\;\;\;\;\;\;\;\;$ divide each side by $4$
$sin(\Theta )=\pm \sqrt{\frac{1}{2}}$
$sin(\Theta )=\pm \frac{1}{\sqrt{2}}$
$\theta= sin^{-1}(\frac{1}{\sqrt{2}})$
We know $ sin(\theta) $ is positive in quadrant $I$ and quadrant $II$
$\theta=45^o\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\theta=180^o-45^o=135^o$
$\theta=45^o\;\;\;\;\;\;\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\;\;\;\theta=135^o$
$\theta= sin^{-1}(\frac{-1}{\sqrt{2}})$
We know $ sin(\theta) $ is negative in quadrant $3$ and quadrant $4$
$\theta=180^o+45^o=225^o\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\theta=360^o-45^o=315^o$
$\theta=225^o\;\;\;\;\;\;\;\;\;\;\;\;\;\;or\;\;\;\;\;\;\;\;\;\;\;\theta=315^o$
$\theta=\{45^o,135^o,225^o,315^o\}$
