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