The value of x satisfying the equation 3tan−1⁡12+3−tan−1⁡1x=tan−1⁡13 is :

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The value of x satisfying the equation $3 \tan^{- 1} \frac{1}{2 + \sqrt{3}} - \tan^{- 1} \frac{1}{x} = \tan^{- 1} \frac{1}{3}$ is :

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$x = \frac{1}{2}$

$x = \frac{1}{2 - \sqrt{3}}$

$x = 2$

None of these

answer is A.

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Detailed Solution

$\begin{array}{l} 3 \tan^{- 1} \frac{1}{2 + \sqrt{3}} - \tan^{- 1} \frac{1}{x} = \tan^{- 1} \frac{1}{3} \\ \Rightarrow 2 \tan^{- 1} \frac{1}{2 + \sqrt{3}} + \tan^{- 1} \frac{1}{2 + \sqrt{3}} - \tan^{- 1} \frac{1}{3} = \tan^{- 1} \frac{1}{x} \\ \Rightarrow \tan^{- 1} \frac{2 \times \frac{1}{2 + \sqrt{3}}}{1 - \frac{1}{(2 + \sqrt{3})^{2}}} + \tan^{- 1} \frac{\frac{1}{2 + \sqrt{3}} - \frac{1}{3}}{1 + \frac{1}{3} \times \frac{1}{2 + \sqrt{3}}} = \tan^{- 1} \frac{1}{x} \\ \Rightarrow \tan^{- 1} \frac{1}{\sqrt{3}} + \tan^{- 1} \frac{1 - \sqrt{3}}{7 + 3 \sqrt{3}} = \tan^{- 1} \frac{1}{x} \\ \Rightarrow \tan^{- 1} \frac{\frac{1}{\sqrt{3}} + \frac{1 - \sqrt{3}}{7 + 3 \sqrt{3}}}{1 - \frac{1}{\sqrt{3}} \times \frac{1 - \sqrt{3}}{7 + 3 \sqrt{3}}} = \tan^{- 1} \frac{1}{x} \\ \Rightarrow \frac{7 + 3 \sqrt{3} + \sqrt{3} - 3}{\sqrt{3} (7 + 3 \sqrt{3}) - (1 - \sqrt{3})} = \frac{1}{x} \Rightarrow x = 2 \end{array}$