cosx + cos2x +cos3x +cos4x +cos5x=0

$\begin{array}{l}
\cos x + \cos 2x + \cos 3x + \cos 4x + \cos 5x = 0\\
 \Leftrightarrow \left( {\cos x + \cos 5x} \right) + \left( {\cos 2x + \cos 4x} \right) + \cos 3x = 0\\
 \Leftrightarrow 2\cos 3x\cos 2x + 2\cos 3x\cos x + \cos 3x = 0\\
 \Leftrightarrow \cos 3x\left( {2\cos 2x + 2\cos x + 1} \right) = 0\\
 \Leftrightarrow \cos 3x\left[ {2\left( {2{{\cos }^2}x - 1} \right) + 2\cos x + 1} \right] = 0\\
 \Leftrightarrow \cos 3x\left( {4{{\cos }^2}x + 2\cos x - 1} \right) = 0\\
 \Leftrightarrow \left[ \begin{array}{l}
\cos 3x = 0\\
4{\cos ^2}x + 2\cos x - 1 = 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\cos 3x = 0\\
\cos x = \dfrac{{ - 1 + \sqrt 5 }}{4}\\
\cos x = \dfrac{{ - 1 - \sqrt 5 }}{4}
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
3x = \dfrac{\pi }{2} + k\pi \\
x =  \pm \arccos \left( {\dfrac{{ - 1 + \sqrt 5 }}{4}} \right) + k2\pi \\
x =  \pm \arccos \left( {\dfrac{{ - 1 - \sqrt 5 }}{4}} \right) + k2\pi 
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{6} + \dfrac{{k\pi }}{3}\\
x =  \pm \arccos \left( {\dfrac{{ - 1 + \sqrt 5 }}{4}} \right) + k2\pi \\
x =  \pm \arccos \left( {\dfrac{{ - 1 - \sqrt 5 }}{4}} \right) + k2\pi 
\end{array} \right.\left( {k \in \mathbb Z} \right)
\end{array}$