Question

The number of distinct real roots of $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$

in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is

• A. $0$ (zero)
• B. $2$
• C. $1$
• D. $>2$

Answer 1

The correct option is C $1$

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

$\Rightarrow {\cos }^{3}x\left|\begin{array}{ccc}\tan x& 1& 1\\ 1& \tan x& 1\\ 1& 1& \tan x\end{array}\right|=0$

$\Rightarrow {\cos }^{3}x(\tan x-1{)}^2(\tan x+2)=0$
$\Rightarrow \cos x=0,\tan x=1,\tan x=-2$

$\cos x\ne 0$ for any $x\in [-\frac{\pi }{4},\frac{\pi }{4}]$
and $\tan x\ne -2$ for any $x\in [-\frac{\pi }{4},\frac{\pi }{4}]$

Thus, it has only one solution $x=\frac{\pi }{4}$ in the given interval.