If `3tan^(-1)(1/(2+sqrt(3)))-tan^(-1)1/x=tan^(-1)1/3,` then `x` is equal to 1 (b) 2 (c) 3 (d) `sqrt(2)`

Explanation

To solve the equation \( 3\tan^{-1}\left(\frac{1}{2+\sqrt{3}}\right) - \tan^{-1}\left(\frac{1}{x}\right) = \tan^{-1}\left(\frac{1}{3}\right) \), we will follow these steps:

### Step 1: Rationalize the term \( \tan^{-1}\left(\frac{1}{2+\sqrt{3}}\right) \)

We rationalize \( \frac{1}{2+\sqrt{3}} \) by multiplying the numerator and denominator by \( 2-\sqrt{3} \):

\[
\frac{1}{2+\sqrt{3}} \cdot \frac{2-\sqrt{3}}{2-\sqrt{3}} = \frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})} = \frac{2-\sqrt{3}}{4 - 3} = 2 - \sqrt{3}
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