$8 \cos x\left\{\cos \left(\frac{\pi}{6}+x\right) \cdot \cos \left(\frac{\pi}{6}-x\right)-\frac{1}{2}\right\}=1$

$\Rightarrow 4 \cos x\left\{2 \cos \left(\frac{\pi}{6}+x\right) \cos \left(\frac{\pi}{6}-x\right)-1\right\}=1$

$\Rightarrow 4 \cos x\left\{\cos 2 x+\cos \frac{\pi}{3}-1\right\}=1$

$\Rightarrow 4 \cos x\left\{\cos 2 x-\frac{1}{2}\right\}=1$

$\Rightarrow 4 \cos x\left\{2 \cos ^{2} x-1-\frac{1}{2}\right\}=1$

$\Rightarrow 8 \cos ^{3} x-6 \cos x=1$

$\Rightarrow 2\left(4 \cos ^{3} x-3 \cos x\right)=1$

$ \cos 3 x=\frac{1}{2}$

$\cos 3x = \cos {60^\circ },\cos {30^\circ },\cos {420^\circ }$

$x{\text{ }} = {20^\circ },{100^\circ },{140^\circ }$

${20^\circ } + {100^\circ } + {140^\circ }{\text{ }} = k\pi $

$ \Rightarrow {260^\circ } = k\left( {{{180}^\circ }} \right)$

$ \Rightarrow k = \frac{{{{260}^\circ }}}{{{{180}^\circ }}} = \frac{{13}}{9}$