$\cos \left( {x + {\pi \over 3}} \right)\cos \left( {{\pi \over 3} - x} \right) = {1 \over 4}{\cos ^2}2x,\,x \in [ - 3\pi ,\,3\pi ]$ $\Rightarrow \cos 2x + \cos {{2\pi } \over 3} = {1 \over 2}{\cos ^2}2x$ $\Rightarrow {\cos ^2}2x - 2\cos 2x - 1 = 0$ $\Rightarrow \cos 2x = 1$ $\therefore$ $x = - 3\pi ,\, - 2\pi ,\, - \pi ,\,0,\,\pi ,\,2\pi ,\,3\pi$ $\therefore$ Number of solutions = 7