Official solution
\[\cos x + \cos 2x + \cos 3x + \cos 4x = 0\]
\[ \Rightarrow \] \[(\cos x + \cos 3x)\] + \[(\cos 2x + \cos 4x)\] = 0
\[ \Rightarrow 2\cos 2x\cos x + 2\cos 3x\cos x = 0\]
\[ \Rightarrow 2\cos x\left( {2\cos {{5x} \over 2}\cos {x \over 2}} \right) = 0\]
\[\cos x = 0,\cos {{5x} \over 2} = 0,\cos {x \over 2} = 0\]
\[x = \pi ,{\pi \over 2},{{3\pi } \over 2},{\pi \over 5},{{3\pi } \over 5},{{7\pi } \over 5},{{9\pi } \over 5}\]