If `tan^(-1) x+tan^(-1) ""1/2 = pi/4 , " then " x=`

Options

A. `-3`
B. 3
C. 2
D. `1//3`

Answer

To solve the equation \( \tan^{-1} x + \tan^{-1} \frac{1}{2} = \frac{\pi}{4} \), we can use the formula for the sum of inverse tangents:

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

where \( ab < 1 \).

### Step 1: Apply the formula
Let \( a = x \) and \( b = \frac{1}{2} \). According to the formula:

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

### Step 2: Set the equation equal to \( \frac{\pi}{4} \)
Since we know that \( \tan^{-1} \frac{\pi}{4} = 1 \), we can set the equation:

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

### Step 3: Take the tangent of both sides
Taking the tangent of both sides gives us:

\[ \frac{x + \frac{1}{2}}{1 - \frac{x}{2}} = 1 \]

### Step 4: Cross-multiply
Cross-multiplying yields:

\[ x + \frac{1}{2} = 1 - \frac{x}{2} \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):

1. Multiply every term by 2 to eliminate the fraction:
\[ 2x + 1 = 2 - x \]

2. Rearranging gives:
\[ 2x + x = 2 - 1 \]
\[ 3x = 1 \]

3. Finally, divide by 3:
\[ x = \frac{1}{3} \]

### Final Answer
Thus, the value of \( x \) is:

\[ \boxed{\frac{1}{3}} \]