Answer
$\frac{\pi}{3},\frac{5\pi}{3}$
Work Step by Step
1. $tan^2\theta=\frac{3}{2}sec\theta \Longrightarrow sec^2\theta-1=\frac{3}{2}sec\theta \Longrightarrow 2sec^2\theta-3sec\theta-2= \Longrightarrow (2sec\theta+1)(sec\theta-2)=0 \Longrightarrow sec\theta=-\frac{1}{2},2$,
2. $sec\theta=-\frac{1}{2} \Longrightarrow $ no solution in $[0,2\pi)$,
3. $sec\theta=2 \Longrightarrow cos\theta=\frac{1}{2} \Longrightarrow \theta=\frac{\pi}{3},\frac{5\pi}{3}$ in $[0,2\pi)$.
