8.084. $\cos 5 x+\cos 7 x=\cos (\pi+6 x)$.

8.084. $\cos 5 x+\cos 7 x=\cos (\pi+6 x)$.
Solution.

From the condition

$2 \cos \frac{5 x+7 x}{2} \cos \frac{5 x-7 x}{2}+\cos 6 x=0, 2 \cos 6 x \cos x+\cos 6 x=0$, $\cos 6 x(2 \cos x+1)=0$.

From this:

1) $\cos 6 x=0, 6 x=\frac{\pi}{2}+\pi n, x_{1}=\frac{\pi}{12}+\frac{\pi n}{6}=\frac{\pi}{12}(2 n+1), n \in Z$;
2) $2 \cos x+1=0, \cos x=-\frac{1}{2}, \quad x_{2}= \pm \frac{2}{3} \pi+2 \pi k, \quad k \in Z$.

Answer: $\quad x_{1}=\frac{\pi}{12}(2 n+1) x_{2}= \pm \frac{2}{3} \pi+2 \pi k, n, k \in Z$.