The system of the equations

                     $x-y\cos \theta +z\cos 2\theta =0$

                     $-x\cos \theta +y-z\cos \theta =0$

                      $x\cos 2\theta -y\cos \theta +z=0$

Has non trivial solution for $\theta$ equals

$\left|\begin{array}{ccc}1& -\cos \theta & \cos 2\theta \\ -\cos \theta & 1& -\cos \theta \\ \cos 2\theta & -\cos \theta & 1\end{array}\right|=0$ 

$Applying{C}_1\to C_1-C_3,then$

$\Rightarrow \left|\begin{array}{ccc}2{\sin }^2\theta & -\cos \theta & \cos 2\theta \\ 0& 1& -\cos \theta \\ -2{\sin }^2\theta & -\cos \theta & 1\end{array}\right|=0$

$\Rightarrow 2{\sin }^2\theta \left|\begin{array}{ccc}1& -\cos \theta & \cos 2\theta \\ 0& 1& -\cos \theta \\ -1& -\cos \theta & 1\end{array}\right|=0$

$\Rightarrow 2{\sin }^2\theta \left\{1(1-{\cos }^2\theta )-1({\cos }^2\theta -\cos 2\theta )\right\}=0$

$\Rightarrow 2{\sin }^2\theta \left\{-2{\cos }^2\theta +1+\cos 2\theta \right\}=0$

$\Rightarrow 2{\sin }^{2}0\left\{0\right\}=0\Rightarrow 0=0$

Which is independent of $\theta$ .