Question

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

• A. $0$
• B. $n$
• C. $2n$
• D. None of the above

Answer 1

The correct option is A $0$

We observe that ${\log }_{\left|\sin x\right|}(1+\cos x)$ is defined
$x\ne n\pi ,\pi (2n+1)\frac{\pi }{2},n\in Z$
Now, ${\log }_{\left|\sin x\right|}(1+\cos x)=2$
$\Rightarrow (1+\cos x)={\left|\sin x\right|}^2$
$\Rightarrow \cos x(1+\cos x)=0$
but ${\cos }^{2}x+\cos x\ne 0$ for any
$x\in (0,n\pi )-(2n-1)\pi /2,n\in Z$
Hence, the given equation has no solution