The value of `cot^(pi)/(4)-2 cot^(-13)` is

Options

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

Answer

To solve the expression \( \cot\left(\frac{\pi}{4}\right) - 2 \cot^{-1}(3) \), we will follow these steps:

### Step 1: Evaluate \( \cot\left(\frac{\pi}{4}\right) \)
The cotangent of \( \frac{\pi}{4} \) is:
\[ \cot\left(\frac{\pi}{4}\right) = 1 \]

### Step 2: Rewrite \( 2 \cot^{-1}(3) \)
Using the identity \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \), we can rewrite \( \cot^{-1}(3) \):
\[ \cot^{-1}(3) = \tan^{-1}\left(\frac{1}{3}\right) \]
Thus,
\[ 2 \cot^{-1}(3) = 2 \tan^{-1}\left(\frac{1}{3}\right) \]

### Step 3: Use the double angle formula for tangent
We can use the formula \( 2 \tan^{-1}(x) = \tan^{-1}\left(\frac{2x}{1-x^2}\right) \). Here, \( x = \frac{1}{3} \):
\[ 2 \tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{2 \cdot \frac{1}{3}}{1 - \left(\frac{1}{3}\right)^2}\right) \]
Calculating the denominator:
\[ 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \]
Now substituting back:
\[ 2 \tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{\frac{2}{3}}{\frac{8}{9}}\right) = \tan^{-1}\left(\frac{2 \cdot 9}{3 \cdot 8}\right) = \tan^{-1}\left(\frac{6}{8}\right) = \tan^{-1}\left(\frac{3}{4}\right) \]

### Step 4: Substitute back into the expression
Now we substitute back into the original expression:
\[ \cot\left(\frac{\pi}{4}\right) - 2 \cot^{-1}(3) = 1 - \tan^{-1}\left(\frac{3}{4}\right) \]

### Step 5: Use the identity for cotangent
Using the identity \( \tan^{-1}(x) = \cot^{-1}\left(\frac{1}{x}\right) \):
\[ 1 - \tan^{-1}\left(\frac{3}{4}\right) = \cot^{-1}\left(\frac{4}{3}\right) \]

### Step 6: Final evaluation
Since \( \cot(\cot^{-1}(x)) = x \), we have:
\[ \cot(\cot^{-1}\left(\frac{4}{3}\right)) = \frac{4}{3} \]

Thus, the final answer is:
\[ \frac{4}{3} \]