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

Question

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

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Answer

$2\tan^{-1}(\cos\text{x})=\tan^{-1}\Bigg(\frac{2\cos\text{x}}{1-\cos^{2}\text{x}}\Bigg)$
$=\tan^{-1}\Bigg(\frac{\text{2 \cos x}}{\text{\sin}^{2}\text{x}}\Bigg)$
$\therefore\tan^{-1}\Bigg(\frac{\text{2 \cos x}}{\text{\sin}^{2}\text{ x}}\Bigg)=\tan^{-1}(2\text{ cosec x})$
$\Rightarrow\frac{2\cos\text{x}}{\sin^{2}\text{x}}=\frac{2}{\text{\sin x}}\Rightarrow\sin\text{x}=\cos\text{x}$
$\Rightarrow\text{x}=\pi/4.$

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