Solve for x: 2 tan-1(cos x) = tan-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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