`2tan^(-1)(1/3)=?`

Options

A. `tan^(-1)2/3`
B. `tan^(-1)3/4`
C. `tan^(-1)4/3`
D. none of these

Explanation

To solve the expression \( 2 \tan^{-1} \left( \frac{1}{3} \right) \), we can use the formula for the double angle of the inverse tangent function:

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

### Step-by-Step Solution:

1. **Identify \( x \)**:
Here, \( x = \frac{1}{3} \).

2. **Apply the formula**:
Substitute \( x \) into the formula:
\[ 2 \tan^{-1} \left( \frac{1}{3} \right) = \tan^{-1} \left( \frac{2 \cdot \frac{1}{3}}{1 - \left( \frac{1}{3} \right)^2} \right) \]

3. **Calculate the numerator**:
The numerator becomes:
\[ 2 \cdot \frac{1}{3} = \frac{2}{3} \]

4. **Calculate the denominator**:
The denominator is:
\[ 1 - \left( \frac{1}{3} \right)^2 = 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]

5. **Combine the results**:
Now, substitute the numerator and denominator into the expression:
\[ 2 \tan^{-1} \left( \frac{1}{3} \right) = \tan^{-1} \left( \frac{\frac{2}{3}}{\frac{8}{9}} \right) \]

6. **Simplify the fraction**:
To simplify:
\[ \frac{\frac{2}{3}}{\frac{8}{9}} = \frac{2}{3} \cdot \frac{9}{8} = \frac{2 \cdot 9}{3 \cdot 8} = \frac{18}{24} = \frac{3}{4} \]

7. **Final result**:
Thus, we have:
\[ 2 \tan^{-1} \left( \frac{1}{3} \right) = \tan^{-1} \left( \frac{3}{4} \right) \]

### Conclusion:
The final answer is:
\[ \boxed{\tan^{-1} \left( \frac{3}{4} \right)} \]