The value of `cot(pi/4-2 cot^(-1)3)` is

Options

A. 1
B. 7
C. -1
D. none of these

Answer

To solve the problem of finding the value of \( \cot\left(\frac{\pi}{4} - 2 \cot^{-1}(3)\right) \), we can follow these steps:

### Step 1: Use the Cotangent Difference Formula
We can use the cotangent difference formula:
\[ \cot(A - B) = \frac{\cot A \cot B + 1}{\cot A + \cot B} \]
In our case, let \( A = \frac{\pi}{4} \) and \( B = 2 \cot^{-1}(3) \).

### Step 2: Calculate \( \cot A \)
Since \( A = \frac{\pi}{4} \):
\[ \cot\left(\frac{\pi}{4}\right) = 1 \]

### Step 3: Calculate \( \cot B \)
To find \( \cot(2 \cot^{-1}(3)) \), we first need to find \( \cot(\cot^{-1}(3)) \):
\[ \cot(\cot^{-1}(3)) = 3 \]
Now, we can use the double angle formula for cotangent:
\[ \cot(2\theta) = \frac{\cot^2(\theta) - 1}{2 \cot(\theta)} \]
Let \( \theta = \cot^{-1}(3) \), then:
\[ \cot(2\theta) = \frac{3^2 - 1}{2 \cdot 3} = \frac{9 - 1}{6} = \frac{8}{6} = \frac{4}{3} \]

### Step 4: Substitute Values into the Cotangent Difference Formula
Now we substitute \( \cot A \) and \( \cot B \) into the cotangent difference formula:
\[ \cot\left(\frac{\pi}{4} - 2 \cot^{-1}(3)\right) = \frac{1 \cdot \frac{4}{3} + 1}{1 + \frac{4}{3}} = \frac{\frac{4}{3} + 1}{1 + \frac{4}{3}} \]

### Step 5: Simplify the Expression
Now simplify the numerator and denominator:
- The numerator:
\[ \frac{4}{3} + 1 = \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \]
- The denominator:
\[ 1 + \frac{4}{3} = \frac{3}{3} + \frac{4}{3} = \frac{7}{3} \]

### Step 6: Final Calculation
Now, substituting back into the expression:
\[ \cot\left(\frac{\pi}{4} - 2 \cot^{-1}(3)\right) = \frac{\frac{7}{3}}{\frac{7}{3}} = 1 \]

Thus, the value of \( \cot\left(\frac{\pi}{4} - 2 \cot^{-1}(3)\right) \) is:
\[ \boxed{1} \]