Question

The number of distinct real values of $x$ in the interval $[-\frac{\pi }{4},\frac{\pi }{4}]$such that $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ is

Answer 1

$\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$
$R_1\to R_1+R_2+R_3\Rightarrow (2\cos x+\sin x)\left|\begin{array}{ccc}1& 1& 1\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0C_1\to C_1-C_3{C}_2\to C_2-C_3\Rightarrow (2\cos x+\sin x)\times \left|\begin{array}{ccc}0& 0& 1\\ 0& \sin x-\cos x& \cos x\\ \cos x-\sin x& \cos x-\sin x& \sin x\end{array}\right|=0$

$\Rightarrow (\sin x-\cos x{)}^2(\sin x+2\cos x)=0$
$\Rightarrow \sin x-\cos x=0$ or $\sin x+2\cos x=0$
$\tan x=1$ or $\tan x=-2$
$x=\frac{\pi }{4}$ or $x={\tan }^{-1}(-2)$
The number of distinct real values of $x$ in the interval $[-\frac{\pi }{4},\frac{\pi }{4}]$ is $1$