What is `tan{2tan^(-1)""(1)/(3)}=`

Options

A. `2//3`
B. `3//4`
C. `3//8`
D. `1//9`

Answer

To solve the problem \( \tan\left(2 \tan^{-1}\left(\frac{1}{3}\right)\right) \), we can use the formula for the tangent of double angles in terms of inverse tangent. The formula we will use is:

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

### Step-by-step Solution:

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

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

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

4. **Calculate the denominator**:
First, calculate \( \left(\frac{1}{3}\right)^2 \):
\[ \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
Now substitute this back into the denominator:
\[ 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]

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

6. **Simplify the fraction**:
To simplify, multiply by the reciprocal of the denominator:
\[ = \frac{2}{3} \cdot \frac{9}{8} = \frac{2 \cdot 9}{3 \cdot 8} = \frac{18}{24} \]

7. **Reduce the fraction**:
Now reduce \( \frac{18}{24} \):
\[ = \frac{3}{4} \]

### Final Answer:
Thus, the value of \( \tan\left(2 \tan^{-1}\left(\frac{1}{3}\right)\right) \) is \( \frac{3}{4} \).

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