A value of $\theta \in \left(0,\frac{\pi }{3}\right)$ for which $\left|\begin{array}{ccc}1+{\cos }^2\theta & {\sin }^2\theta & 4\cos 6\theta \\ {\cos }^2\theta & 1+{\sin }^2\theta & 4\cos 6\theta \\ {\cos }^2\theta & {\sin }^2\theta & 1+4\cos 6\theta \end{array}\right|=0$ is

We have

$\begin{array}{l}{R}_2\to R_2-R_1, R_3\to R_3-R_2\\ \Rightarrow \left|\begin{array}{ccc}1+{\cos }^2\theta & {\sin }^2\theta & 4\cos 6\theta \\ -1& 1& 0\\ -1& 0& 1\end{array}\right|=0\\ \Rightarrow \left(1+{\cos }^2\theta \right)·1-{\sin }^2\theta (-1)+4\cos 6\theta =0\\ \Rightarrow 2+4\cos 6\theta =0\\ \Rightarrow \cos 6\theta =\frac{-1}{2}\\ \Rightarrow 6\theta =\frac{2\pi }{3},\frac{4\pi }{3}\\ \Rightarrow \theta =\frac{\pi }{9},\frac{2\pi }{9}\end{array}$