Solve for x: 2 ^ -1 ( x) = ^ -1 (2cosec x) - Vidyadip

Question

$\text{Solve for x:}$
$2\tan^{-1}(\cos x) = \tan^{-1}(2\text{cosec x)} $

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Answer

$2\tan^{-1}(\cos\text{x}) = \tan^{-1}(2\text{cosec x)}$
$\Rightarrow\tan^{-1}\bigg(\frac{2 \cos \text{x}}{1 - \cos^{2}\text{x}}\bigg)= \tan^{-1}\bigg(\frac{2}{\sin \text{x}}\bigg)$
$\Rightarrow \sin\text{x}(\sin\text{x} - \cos \text{x}) = 0$
$\Rightarrow\sin\text{x} = \cos\text{x}$
the solution is $\text{x} = \frac{\pi}{4}$

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