Question

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

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

Answer 1

The correct option is B a unique solution

We have, $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$
$\therefore$ $–1\le \sin \theta \le 1,$ here $0<\sin \theta <1$
Now, ${\log }_{\sin \theta }\cos 2\theta =2$
$\Rightarrow \cos 2\theta ={\sin }^2\theta \Rightarrow 1-2{\sin }^2\theta ={\sin }^2\theta$
$\Rightarrow 3{\sin }^2\theta =1\Rightarrow {\sin }^2\theta =\frac{1}{3}\Rightarrow \sin \theta =\frac{1}{\sqrt{3}}$ $(\because 0<\sin \theta <1)$
The given equation has unique solution.