Question

In the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ the equation ${\log }_{\sin }\left(\cos 2\theta \right)=2$ has

• A. no solution
• B. a unique solution
• C. two solutions
• D. infinitely many solutions

Answer 1

The correct option is B a unique solution

$\therefore -\frac{\pi }{2}\le \theta \le \frac{\pi }{2}\Rightarrow -1\le \sin \theta \le 1$
Here $0<\sin \theta <1\Rightarrow {\log }_{\sin \theta }\cos 2\theta =2$
$\cos 2\theta ={\sin }^2\theta \Rightarrow {\log }_{\sin \theta }\cos 2\theta =2$
$3{\sin }^2\theta =1\Rightarrow {\sin }^2\theta =\frac{1}{3}$
$\therefore \sin \theta =\frac{1}{\sqrt{3}}\left\{\therefore 0<\sin \theta <1\right\}$ a unique solution