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

Options

A. `-tan^(-1)((4)/(3))`
B. `-tan^(-1)((3)/(4))`
C. `tan^(-1)((4)/(3))`
D. `tan^(-1)((3)/(4))`

Answer

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

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

### Step 1: Let \( y = \tan^{-1}\left(\frac{1}{3}\right) \)

This means that \( \tan(y) = \frac{1}{3} \).

### Step 2: Apply the double angle formula

Using the double angle formula for tangent, we have:

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

### Step 3: Substitute \( \tan(y) \)

Substituting \( \tan(y) = \frac{1}{3} \) into the formula:

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

### Step 4: Calculate \( \tan(2y) \)

First, calculate \( \left(\frac{1}{3}\right)^2 = \frac{1}{9} \).

Now substitute this value into the equation:

\[ \tan(2y) = \frac{\frac{2}{3}}{1 - \frac{1}{9}} = \frac{\frac{2}{3}}{\frac{8}{9}} \]

### Step 5: Simplify the fraction

To simplify \( \frac{\frac{2}{3}}{\frac{8}{9}} \):

\[ \tan(2y) = \frac{2}{3} \cdot \frac{9}{8} = \frac{2 \cdot 9}{3 \cdot 8} = \frac{18}{24} = \frac{3}{4} \]

### Step 6: Find \( 2y \)

Now, we have:

\[ \tan(2y) = \frac{3}{4} \]

Thus, we can write:

\[ 2y = \tan^{-1}\left(\frac{3}{4}\right) \]

### Step 7: Conclusion

Therefore, we conclude that:

\[ 2\tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{3}{4}\right) \]

So, the final answer is:

\[ \boxed{\tan^{-1}\left(\frac{3}{4}\right)} \]