The number of solutions of the equation $\cos (x+\frac{\pi }{3})\cos (\frac{\pi }{3}-x)=\frac{1}{4}{\cos }^{2}2xinx\in [-3\pi ,3\pi ]$  is 

${\cos }^2\frac{\pi }{3}-{\sin }^{2}x=\frac{1}{4}{(1-2{\sin }^{2}x)}^2$

$\frac{1}{4}-{\sin }^{2}x=\frac{1}{4}(1+4{\sin }^{4}x-4{\sin }^{2}x)$

$\frac{1}{4}-{\sin }^{2}x=\frac{1}{4}+{\sin }^{4}x-{\sin }^{2}x$

${\sin }^{4}x=0\Rightarrow \sin x=0=-3\pi ,-2\pi ,-\pi ,0,\pi ,2\pi ,3\pi$

$\therefore Number of solutions = 7$