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