Question

The sum of all the solution of the equation $\cot \theta =\sin 2\theta (\theta \ne n\pi ,n$ integer, $0\le \theta \le \pi )$ is

• A. $\frac{3\pi }{2}$
• B. $\pi$
• C. $\frac{3\pi }{4}$
• D. $2\pi$

Answer 1

Answer: (A)

$\frac{3\pi }{2}$

From the given relation

$\cos \theta =\left(2\sin \theta \cos \theta \right)\sin \theta \ne 0$

$\Rightarrow \sin \theta =\pm \frac{1}{\sqrt{2}}$ or $\cos \theta =0$

$\Rightarrow \theta =\frac{\pi }{4},\frac{3\pi }{4},\frac{\pi }{2}$

Then the sum of roots is $\frac{3\pi }{2}$