Answer
$\frac{5\pi}{3}$
Work Step by Step
1. $cot\theta+csc\theta=-\sqrt 3 \Longrightarrow cot\theta+\sqrt 3=-csc\theta \Longrightarrow cot^2\theta+2\sqrt 3cot\theta+3=csc^2\theta=1+cot^2\theta \Longrightarrow cot\theta=-\frac{\sqrt 3}{3}$
2. $\theta=k\pi+\frac{2\pi}{3}$
3. Within $[0,2\pi)$, we have $\theta=\frac{2\pi}{3},\frac{5\pi}{3}$, check only $\theta=\frac{5\pi}{3}$ fits the original equation.
