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

Options

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

Answer

To solve the equation \( \cot^{-1} x + \tan^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{4} \), we can follow these steps:

### Step 1: Rewrite \( \cot^{-1} x \)
We know that:
\[ \cot^{-1} x = \tan^{-1} \left( \frac{1}{x} \right) \]
So, we can rewrite the equation as:
\[ \tan^{-1} \left( \frac{1}{x} \right) + \tan^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{4} \]

### Step 2: Use the formula for the sum of inverse tangents
We can use the formula for the sum of two inverse tangents:
\[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \]
provided that \( ab < 1 \). Here, let \( a = \frac{1}{x} \) and \( b = \frac{1}{2} \). Thus, we have:
\[ \tan^{-1} \left( \frac{\frac{1}{x} + \frac{1}{2}}{1 - \frac{1}{x} \cdot \frac{1}{2}} \right) = \frac{\pi}{4} \]

### Step 3: Set the equation equal to 1
Since \( \tan \left( \frac{\pi}{4} \right) = 1 \), we can equate:
\[ \frac{\frac{1}{x} + \frac{1}{2}}{1 - \frac{1}{2x}} = 1 \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ \frac{1}{x} + \frac{1}{2} = 1 - \frac{1}{2x} \]

### Step 5: Clear the fractions
To eliminate the fractions, multiply through by \( 2x \):
\[ 2 + x = 2x - 1 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 2 + x + 1 = 2x \]
\[ 3 = 2x - x \]
\[ 3 = x \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{3} \]