Question

Solve $\cot \theta +\tan \theta =2\csc \theta$

Answer 1

Given, $\cot \theta +\tan \theta =2\csc \theta$
$\Rightarrow \frac{\cos \theta }{\sin \theta }+\frac{\sin \theta }{\cos \theta }=\frac{2}{\sin \theta }$
$\Rightarrow \frac{{\cos }^2\theta +{\sin }^2\theta }{\sin \theta \cos \theta }=\frac{2}{\sin \theta }$
$\Rightarrow \frac{1}{\sin \theta \cos \theta }=\frac{2}{\sin \theta }$
$\Rightarrow 2\cos \theta =1$

$\Rightarrow \cos \theta =\frac{1}{2}$

$\Rightarrow \theta =2n\pi \pm \frac{\pi }{3}$