Apply: $R_{1} \rightarrow R_{1}-R_{2}$ and $R_{2} \rightarrow R_{2}-R_{3}$

$\left|\begin{array}{ccc} \sin x-\cos x & \cos x-\sin x & 0 \\ 0 & \sin x-\cos x & \cos x-\sin x \\ \cos x & \cos x & \sin x \end{array}\right|=0$

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

$(\sin x-\cos x)^{2}(\sin x+2 \cos x)=0$

$\sin x=\cos x \quad$ or $\quad \sin x=-2 \cos x$

$\tan x=1$ or $\quad \tan x=-2$  (Not valid)

$\therefore x=\frac{\pi}{4}$