Question

If $cot\Theta =sin2\Theta (where\Theta \ne n\pi ,nisaninteger)\Theta$=

• A. 45${}^0$ and 60${}^0$
• B. 45 ${}^0$ and 90 ${}^0$
• C. only 45${}^0$
• D. only 90${}^0$

Answer 1

The correct option is A 45${}^0$ and 60${}^0$

$\cot \theta =\sin 2\theta$

$\Rightarrow \frac{\cos \theta }{\sin \theta }=2\sin \theta \cos \theta$

$\Rightarrow \cos \theta =2{\sin }^2\theta \cos \theta$

$\Rightarrow \cos \theta -2{\sin }^2\theta \cos \theta =0$

$\Rightarrow \cos \theta (1-2{\sin }^2\theta )=0$

$\Rightarrow \cos \theta \left(\cos 2\theta \right)=0$

$\Rightarrow \cos \theta =0$ or $\cos 2\theta =0$

$\Rightarrow \theta =\frac{\pi }{2}$ or $2\theta =\frac{\pi }{2}$

$\Rightarrow \theta =\frac{\pi }{4}$

$\therefore \theta$ can be ${45}^o$ or ${90}^o$.