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,
$\tan ^{\prime}\left(\frac{2 \cos \theta}{1-\cos ^{2} \theta}\right)=\tan ^{\prime}(2 \operatorname{cosec} \theta) \quad\left(\because 2 \tan ^{-1} x=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\right)$
$\frac{2 \cos \theta}{\sin ^{2} \theta}=2 \operatorname{cosec} \theta$
$\frac{2 \cos \theta}{\sin ^{2} \theta}=\frac{2}{\sin \theta}$
$\frac{\cos \theta}{\sin \theta}=1 \quad \Rightarrow \quad \cot \theta=1 \quad \Rightarrow \quad \theta=\frac{\pi}{4}$