$\theta  \in \left( {0,\frac{\pi }{3}} \right)$

$\left| {\begin{array}{*{20}{c}}
{1 + {{\cos }^2}\theta }&{{{\sin }^2}\theta }&{4\cos 6\theta }\\
{{{\cos }^2}\theta }&{1 + {{\sin }^2}\theta }&{4\cos 6\theta }\\
{{{\cos }^2}\theta }&{{{\sin }^2}\theta }&{1 + 4\cos 6\theta }
\end{array}} \right| = 0$

${R_2} \to {R_2} – {R_1},{R_3} \to {R_3} – {R_1}$

$ \Rightarrow \left| {\begin{array}{*{20}{c}}
{1 + {{\cos }^2}\theta }&{{{\sin }^2}\theta }&{4\cos 6\theta }\\
{ – 1}&1&0\\
{ – 1}&0&1
\end{array}} \right| = 0$

${C_1} \to {C_1} + {C_2}$

$ \Rightarrow \left| {\begin{array}{*{20}{c}}
2&{{{\sin }^2}\theta }&{4\cos 6\theta }\\
0&1&0\\
{ – 1}&0&1
\end{array}} \right| = 0$

expanding along first column

$ \Rightarrow 2\left[ {1 – 0} \right] – 1\left[ { – 4\cos 6\theta } \right] = 0$

$ \Rightarrow 2 + 4\cos 6\theta  = 0$

$ \Rightarrow \cos 6\theta  =  – \frac{1}{2}$

$ \Rightarrow 6\theta  = \frac{{2\pi }}{3}$

$ \Rightarrow \theta  = \frac{\pi }{9}$