The number of distinct real roots of | cases x & x & x x & x & x x...

We have, \left|\begin{cases}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{cases}\right| = 0, - \dfrac{\pi}{4} \le x \le \dfrac{\pi}{4}

Apply: ${R}_{1} \to {R}_{1} - {R}_{2}$ and ${R}_{2} \to {R}_{2} - {R}_{3}$

\left|$\begin{cases}\sin x - \cos x & \cos x - \sin x & 0 \\ 0 & \sin x - \cos x & \cos x - \sin x \\ \cos x & \cos x & \sin x\end{cases}$\right| = 0$</p><p>$⇒ {\left(\sin x - \cos x\right)}^{2}\left|$\begin{cases}1 & - 1 & 0 \\ 0 & 1 & - 1 \\ \cos x & \cos x & \sin x\end{cases}$\right| = 0$</p><p>$⇒ {\left(\sin x - \cos x\right)}^{2}\left(\sin x + 2\cos x\right) = 0$</p><p>$⇒ \sin x = \cos x or \sin x = - 2\cos x$</p><p>$∴ x = \dfrac{\pi}{4}$</p><p>Since,$\dfrac{- \pi}{4} \le x \le \dfrac{\pi}{4},$we get$- 1 \le \tan x \le 1$</p><p>$⇒ \tan x = 1 ⇒ x = \dfrac{\pi}{4}

Hence, number of roots is one.