`cos(x + pi/6) + cos 2x = 0`

`<=> cos ( x + pi/6 ) = - cos 2x`

`<=>  cos ( x + pi/6 ) =  cos ( pi - 2x)`

`<=>` $\left[\begin{matrix} x+\dfrac{\pi}{6}=\pi-2x+k2\pi\\x+\dfrac{\pi}{6}=-\pi+2x+k2\pi\end{matrix}\right.$

`<=>` $\left[\begin{matrix} 3x=\dfrac{5\pi}{6}+k2\pi\\-x=-\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.$

`<=>`  $\left[\begin{matrix} x=\dfrac{5\pi}{18}+\dfrac{k2\pi}{3}\\x=\dfrac{7\pi}{6}-k2\pi\end{matrix}\right.$ `( k in ZZ)`