Complete set of values of x in  $(0,\pi )$  satisfying the inequality  $1+{\log }_2(-\cos x)+{\log }_2\cos 3x\ge 0$, is –

 $1+{\log }_2(-\cos x)+{\log }_2\cos 3x\ge 0$  … (1)
Necessary condition
$\begin{array}{c}\cos x<0\\ \cos 3x>0\end{array}\}n\Rightarrow x\in (\frac{\pi }{2},\frac{5\pi }{6})$
Now, (1)  $\Rightarrow -2\cos x.\cos 3x\ge 1\Rightarrow 8\cos 4x-6{\cos }^{2}x+1\le 0$
 $\Rightarrow (4{\cos }^{2}x-1)(2{\cos }^{2}x-1)\le 0\Rightarrow -\frac{1}{\sqrt{2}}\le \cos x\le -\frac{1}{2}\Rightarrow x\in [\frac{2\pi }{3},\frac{3\pi }{4}]$