The number of distinct real roots of sin xcos xcos xcos xsin xcos xcos xcos xsin x=0 in the interal -π4≤x≤π4is:

C1→C1+C2+C3 (x cos x+sin x) 1 cos xcos x1sin xcos x1cos xsin x=0 (2cos x+sin x) 0 cos x-sin x00-(cos x-sin x)cos x-sin x1cos xsin x=0 R1→R1-R3 R2→R2-R3 ⇒ (2cos x+sin x) (cos x-sin x)2=0 ⇒ 2cos x+sin x=0, (or) cos x-sin x=0 ⇒ tan x≠-2 ⇒ cos x=sin x tan x=1 x=π4

The number of distinct real roots of sin xcos xcos xcos xsin xcos xcos xcos xsin x=0 in the interal -π4≤x≤π4is:

C1→C1+C2+C3 (x cos x+sin x) 1 cos xcos x1sin xcos x1cos xsin x=0 (2cos x+sin x) 0 cos x-sin x00-(cos x-sin x)cos x-sin x1cos xsin x=0 R1→R1-R3 R2→R2-R3 ⇒ (2cos x+sin x) (cos x-sin x)2=0 ⇒ 2cos x+sin x=0, (or) cos x-sin x=0 ⇒ tan x≠-2 ⇒ cos x=sin x tan x=1 x=π4

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The number of distinct real roots of $|\begin{matrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{matrix}| = 0 in the interal - \frac{π}{4} \leq x \leq \frac{π}{4}$ is:

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Detailed Solution

$C_{1} \to C_{1} + C_{2} + C_{3} (x \cos x + \sin x) |\begin{matrix} 1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x \end{matrix}| = 0 (2 \cos x + \sin x) |\begin{matrix} 0 & \cos x - \sin x & 0 \\ 0 & - (\cos x - \sin x) & \cos x - \sin x \\ 1 & \cos x & \sin x \end{matrix}| = 0 R_{1} \to R_{1} - R_{3} R_{2} \to R_{2} - R_{3}$

$\Rightarrow (2 \cos x + \sin x) {(\cos x - \sin x)}^{2} = 0 \Rightarrow 2 \cos x + \sin x = 0, (or) \cos x - \sin x = 0 \Rightarrow \tan x \neq - 2 \Rightarrow \cos x = \sin x \tan x = 1 x = \frac{π}{4}$