Write the solution set of the equation

Given: $(2 \cos x+1)(4 \cos x+5)=0$

Now, $2 \cos x+1=0$ or $4 \cos x+5=0$

$\Rightarrow \cos x=-\frac{1}{2}$ or $\cos x=-\frac{5}{4}$

$\cos x=-\frac{5}{4}$ is not possible.

Thus, we have:

$\cos x=-\frac{1}{2}$

$\Rightarrow \cos x=\cos \frac{2 \pi}{3}$

$\Rightarrow x=2 n \pi \pm \frac{2 \pi}{3}$

By putting n = 0 and n = 1 in the above equation, we get:

$x=\frac{2 \pi}{3}$ or $x=\frac{4 \pi}{3}$ in the interval $[0,2 \pi]$

For the other value of n, x will not satisfy the given condition.

$\therefore x=\frac{2 \pi}{3}$ and $\frac{4 \pi}{3}$