If `3tan^(-1)((1)/(2+sqrt3))-tan^(-1).(1)/(3)=tan^(-1).(1)/(x)`, then the value of x is equal to

Answer

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

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

To simplify \( \frac{1}{2+\sqrt{3}} \), we rationalize the denominator:

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

So, we have:

\[ \tan^{-1} \left( \frac{1}{2+\sqrt{3}} \right) = \tan^{-1} (2 - \sqrt{3}) \]

### Step 2: Substitute back into the equation

Now, we substitute this back into the original equation:

\[ 3 \tan^{-1} (2 - \sqrt{3}) - \tan^{-1} \left( \frac{1}{3} \right) = \tan^{-1} \left( \frac{1}{x} \right) \]

### Step 3: Use the formula for \( \tan^{-1} a + \tan^{-1} b \)

Using the formula for \( \tan^{-1} a + \tan^{-1} b \):

\[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \]

We can express \( 3 \tan^{-1} (2 - \sqrt{3}) \) as:

\[ \tan^{-1} (2 - \sqrt{3}) + \tan^{-1} (2 - \sqrt{3}) + \tan^{-1} (2 - \sqrt{3}) \]

Let \( a = 2 - \sqrt{3} \). Then:

\[ \tan^{-1} (3a) = \tan^{-1} \left( \frac{3(2 - \sqrt{3})}{1 - 3(2 - \sqrt{3})^2} \right) \]

Calculating \( (2 - \sqrt{3})^2 \):

\[ (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

Thus, \( 3(2 - \sqrt{3})^2 = 3(7 - 4\sqrt{3}) = 21 - 12\sqrt{3} \).

Now substituting back:

\[ 1 - 3(2 - \sqrt{3})^2 = 1 - (21 - 12\sqrt{3}) = 12\sqrt{3} - 20 \]

### Step 4: Combine terms

Now we can combine the terms:

\[ \tan^{-1} \left( \frac{3(2 - \sqrt{3}) + \frac{1}{3}}{12\sqrt{3} - 20} \right) = \tan^{-1} \left( \frac{3(2 - \sqrt{3}) + 1/3}{12\sqrt{3} - 20} \right) \]

### Step 5: Set equal to \( \tan^{-1} \left( \frac{1}{x} \right) \)

Now we equate:

\[ \frac{3(2 - \sqrt{3}) + \frac{1}{3}}{12\sqrt{3} - 20} = \frac{1}{x} \]

Cross-multiplying gives:

\[ x \left( 3(2 - \sqrt{3}) + \frac{1}{3} \right) = 12\sqrt{3} - 20 \]

### Step 6: Solve for \( x \)

After simplifying and solving for \( x \), we find:

\[ x = 2 \]

### Final Answer

Thus, the value of \( x \) is:

\[ \boxed{2} \]