Answer
$\frac{3\pi}{4},\frac{7\pi}{4}$
Work Step by Step
1. $(cot\theta+1)(csc\theta-\frac{1}{2})=0 \Longrightarrow cot\theta=-1\ (tan\theta=-1)\ \\or\ csc\theta=\frac{1}{2}\ (sin\theta=2)$,
2. $sin\theta=2 \Longrightarrow$ no solution in $[0,2\pi)$,
3. $tan\theta=-1 \Longrightarrow \theta=\frac{3\pi}{4},\frac{7\pi}{4}$ in $[0,2\pi)$.
