Answer
$$x=\frac{\pi }{3}, \qquad x=\frac{2\pi }{3}, \\ x=\frac{4\pi }{3}, \qquad x=\frac{5\pi }{3}$$
Work Step by Step
We solve for the angles as follows:
$$4\cos ^2 x -1=0 \quad \Rightarrow \quad \cos ^2 \theta = \frac{1}{4}\quad \Rightarrow \quad \cos \theta = \pm \frac{1}{2} \quad \Rightarrow \quad x=\frac{\pi }{3}+ 2n\pi, \quad \text{ or } \quad x=\frac{2\pi }{3} + 2n \pi, \quad \text{ or } \quad x=\frac{4\pi }{3}+2n \pi, \quad \text{ or } \quad x=\frac{5\pi }{3}+2n\pi \quad n \in \mathbb{Z}$$So, $x= \frac{\pi }{3}$, $x= \frac{2\pi }{3}$, $x=\frac{4\pi }{3}$, and $x=\frac{5\pi }{3}$ are the only solutions in the interval $[0, 2\pi )$.