If 2 ^ -1 ( )= ^ -1 (2cosec ), then show that = 4 , where n is an… - Vidyadip

Question

If $2\tan^{-1}(\cos\theta)=\tan^{-1}(2\text{cosec}\theta),$ then show that $\theta=\frac{\pi}{4},$ where n is any integer.

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Answer

We have, $2\tan^{-1}(\cos\theta)=\tan^{-1}(2\text{cosec}\theta)$
$\Rightarrow\ \tan^{-1}\Big(\frac{2\cos\theta}{1-\cos^2\theta}\Big)=\tan^{-1}(2\text{cosec}\theta)$
$\Big[\because\ 2\tan^{-1}\text{x}=\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)\Big]$
$\Rightarrow\ \frac{2\cos\theta}{\sin^2\theta}=(2\text{cosec}\theta)$
$\Rightarrow\ \cot\theta.2\text{ cosec}\theta=2\text{ cosec}\theta$
$\Rightarrow\ \cot\theta=1$
$\Rightarrow\ \cot\theta=\cot\frac{\pi}{4}$
$\Rightarrow\ \theta=\frac{\pi}{4}$

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