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If $\cot \frac{2 x}{3}+\tan \frac{x}{3}=\operatorname{cosec} \frac{k x}{3},$ then the value of $k$ is
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Given, $\cot \frac{2 x}{3}+\tan \frac{x}{3}=\operatorname{cosec} \frac{k x}{3}$
Let $\theta=\frac{x}{3},$ then
$\therefore \quad \cot 2 \theta+\tan \theta=\operatorname{cosec} k \theta$
$\Rightarrow \quad \frac{\cos 2 \theta}{\sin 2 \theta}+\frac{\sin \theta}{\cos \theta}=\operatorname{cosec} k \theta$
$\Rightarrow \frac{2 \cos ^{2} \theta-1}{2 \sin \theta \cos \theta}+\frac{\sin \theta}{\cos \theta}=\operatorname{cosec} k \theta$
$\Rightarrow \frac{2 \cos ^{2} \theta-1+2 \sin ^{2} \theta}{2 \sin \theta \cos \theta}=\operatorname{cosec} k \theta$
$\Rightarrow \quad \frac{1}{2 \sin \theta \cos \theta}=\operatorname{cosec} k \theta$
$\Rightarrow \quad \operatorname{cosec} 2 \theta=\operatorname{cosec} k \theta$
$\Rightarrow \quad k=2$
Let $\theta=\frac{x}{3},$ then
$\therefore \quad \cot 2 \theta+\tan \theta=\operatorname{cosec} k \theta$
$\Rightarrow \quad \frac{\cos 2 \theta}{\sin 2 \theta}+\frac{\sin \theta}{\cos \theta}=\operatorname{cosec} k \theta$
$\Rightarrow \frac{2 \cos ^{2} \theta-1}{2 \sin \theta \cos \theta}+\frac{\sin \theta}{\cos \theta}=\operatorname{cosec} k \theta$
$\Rightarrow \frac{2 \cos ^{2} \theta-1+2 \sin ^{2} \theta}{2 \sin \theta \cos \theta}=\operatorname{cosec} k \theta$
$\Rightarrow \quad \frac{1}{2 \sin \theta \cos \theta}=\operatorname{cosec} k \theta$
$\Rightarrow \quad \operatorname{cosec} 2 \theta=\operatorname{cosec} k \theta$
$\Rightarrow \quad k=2$
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