Question

The number of distinct solutions of the equation, ${\log }_{\frac{1}{2}}|\sin x|=2-{\log }_{\frac{1}{2}}|\cos x|$ in the interval $[0,2\pi ],$ is

Answer 1

${\log }_{\frac{1}{2}}|\sin x|=2-{\log }_{\frac{1}{2}}|\cos x|,x\in [0,2\pi ]$
$\Rightarrow {\log }_{\frac{1}{2}}|\sin x||\cos x|=2$
$\Rightarrow |\sin x\cos x|=\frac{1}{4}$
$\therefore \sin 2x=\pm \frac{1}{2}$


$\therefore$ We have 8 solutions for $x\in [0,2\pi ]$