Answer
$\frac{2\pi}{3},\frac{4\pi}{3}$
Work Step by Step
Using trigonometric identities, we have
$4sec\theta+6=-2 \longrightarrow \frac{4}{cos\theta}=-8 \longrightarrow cos\theta=-\frac{1}{2} \longrightarrow \theta=\pi-\frac{\pi}{3},\pi+\frac{\pi}{3} \longrightarrow \theta=\frac{2\pi}{3},\frac{4\pi}{3}$
