$\left|\begin{array}{ccc}1+{\sin }^2\mathrm{\theta }& {\sin }^2\mathrm{\theta }& {\sin }^2\mathrm{\theta }\\ {\cos }^2\mathrm{\theta }& 1+{\cos }^2\mathrm{\theta }& {\cos }^2\mathrm{\theta }\\ 4\sin 4\mathrm{\theta }& 4\sin 4\mathrm{\theta }& 1+4\sin 4\mathrm{\theta }\end{array}\right|=0$ then sin $4\mathrm{\theta }$ equals to

Let $\mathrm{A}\equiv \left|\begin{array}{ccc}1+{\sin }^2\mathrm{\theta }& {\sin }^2\mathrm{\theta }& {\sin }^2\mathrm{\theta }\\ {\cos }^2\mathrm{\theta }& 1+{\cos }^2\mathrm{\theta }& {\cos }^2\mathrm{\theta }\\ 4\sin 4\mathrm{\theta }& 4\sin 4\mathrm{\theta }& 1+4\sin 4\mathrm{\theta }\end{array}\right|=0$
Applying ${\mathrm{R}}_1\to {\mathrm{R}}_1+{\mathrm{R}}_2$
$\left|\begin{array}{ccc}2& 2& 1\\ {\cos }^2\mathrm{\theta }& 1+{\cos }^2\mathrm{\theta }& {\cos }^2\mathrm{\theta }\\ 4\sin 4\mathrm{\theta }& 4\sin 4\mathrm{\theta }& 1+4\sin 4\mathrm{\theta }\end{array}\right|=0$
Applying${\mathrm{C}}_1\to {\mathrm{C}}_1-2{\mathrm{C}}_3,{\mathrm{C}}_2\to {\mathrm{C}}_2-2{\mathrm{C}}_3$

$$\begin{gathered} \left|\begin{array}{ccc}1& 1& 1\\ 0& 1& {\cos }^2\mathrm{\theta }\\ -1& -1& 1+4\sin 4\mathrm{\theta }\end{array}\right|=0 \\ {C}_1\to C_1-C_2 \\ \left|\begin{array}{ccc}0& 1& 1\\ -1& 1& {\cos }^2\mathrm{\theta }\\ 0& -1& 1+4\sin 4\mathrm{\theta }\end{array}\right|=0 \\ Determinant expanding along C_1 then \\ -1\left(1+4\sin 4\theta +1\right)=0 \\ 4\sin 4\theta =-2 \\ \sin 4\theta =\frac{-1}{2} \end{gathered}$$