Answer
$\frac{\pi}{2},\frac{3\pi}{2},\frac{\pi}{4},\frac{5\pi}{4}$
Work Step by Step
1. $csc^2\theta=cot\theta+1 \Longrightarrow 1+cot^2\theta=cot\theta+1 \Longrightarrow cot^2\theta-cot\theta=0 \Longrightarrow cot\theta(cot\theta-1)=0 \Longrightarrow cot\theta=0,1$,
2. $cot\theta=0 \Longrightarrow \theta=\frac{\pi}{2},\frac{3\pi}{2}$ in $[0,2\pi)$,
3. $cot\theta=1 \Longrightarrow \theta=\frac{\pi}{4},\frac{5\pi}{4}$ in $[0,2\pi)$.
