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If 2 tan-1(cos θ) = tan-1 (2 cosec θ),

Question:

 If 2 tan-1(cos θ) = tan-1 (2 cosec θ), then show that θ = π/4.

Solution:

Given, 2 tan-1(cos θ) = tan-1 (2 cosec θ)

So,

$\tan ^{\prime}\left(\frac{2 \cos \theta}{1-\cos ^{2} \theta}\right)=\tan ^{\prime}(2 \operatorname{cosec} \theta) \quad\left(\because 2 \tan ^{-1} x=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\right)$

$\frac{2 \cos \theta}{\sin ^{2} \theta}=2 \operatorname{cosec} \theta$

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

$\frac{\cos \theta}{\sin \theta}=1 \quad \Rightarrow \quad \cot \theta=1 \quad \Rightarrow \quad \theta=\frac{\pi}{4}$

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