Question

The number of values of $x\in [0,\pi ]$, that satisfies the equation ${\log }_{|\sin x|}(1+\cos x)=2$, is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

For $\log$ to be defined,
$|\sin x|\ne 1,1+\cos x>0\forall x\in [0,\pi ]$
$\Rightarrow x\ne \frac{\pi }{2},\pi$

Now, ${\log }_{|\sin x|}(1+\cos x)=2$
$\Rightarrow 1+\cos x={\sin }^{2}x$
$\Rightarrow 1+\cos x=1-{\cos }^{2}x$
$\Rightarrow \cos x(1+\cos x)=0$
$\Rightarrow \cos x=0$ or $\cos x=-1$
$\Rightarrow x=\frac{\pi }{2}$ or $x=\pi$
But $x\ne \frac{\pi }{2},\pi$

Hence, there is no solution in $[0,\pi ]$.