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If $\cot \theta + \tan \theta = 2 cosec \theta$ , then find the general value of $\theta$ .

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$\text{Given that}\cot \theta +tan \theta =2 cosec \theta ~~~ \\\\$

$\\ ~~\frac{\cos \theta }{\sin \theta }+\frac{\sin \theta }{\cos \theta }=\frac{2}{\sin \theta }~~ \\\\ ~~~~\frac{\\sin ^{2} \theta +\\cos ^{2} \theta }{\sin \theta \cos \theta }=\frac{2}{\sin \theta }~~~~~~ \\\\ ~~~~~\frac{1}{\sin \theta \cos \theta }=\frac{2}{\sin \theta }~~~~~~~ \\\\ ~2\sin \theta \cos \theta =\sin \theta \\\\$
$\\ ~ \sin \theta \left( 2\cos \theta - 1 \right) =0 \\\\ ~\sin \theta =0~or\: 2\cos \theta - 1=0 \: or\: \cos \theta =\frac{1}{2}~~~~~~ \\\\ ~~Now,~\sin \theta =0\Rightarrow \theta =n \pi , n \epsilon Z \\\\ ~ \cos \theta =\frac{1}{2}\Rightarrow \cos \theta =\cos \frac{ \pi }{3}~~ \\\\ ~ \theta =2n \pi \pm \frac{ \pi }{3}~ \\\\$

Hence, general values of $\theta$ is $2n \pi \pm \frac{ \pi }{3}~ and \: \: n \pi ,n \epsilon Z \\\\$

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