The value of `cot((pi)/(4)+theta)cot((pi)/(4)-theta)` is

Options

A. -1
B. 0
C. 1
D. Not defined

Explanation

To find the value of \( \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) \), we can use the trigonometric identities for cotangent of sum and difference.

### Step-by-Step Solution:

1. **Identify the expression**:
\[ \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) \]

2. **Use the cotangent addition formula**:
The formula for \( \cot(a + b) \) is:
\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]
Here, let \( a = \frac{\pi}{4} \) and \( b = \theta \).

Therefore,
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{\cot\left(\frac{\pi}{4}\right) \cot(\theta) - 1}{\cot\left(\frac{\pi}{4}\right) + \cot(\theta)} \]

3. **Substitute \( \cot\left(\frac{\pi}{4}\right) = 1 \)**:
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{1 \cdot \cot(\theta) - 1}{1 + \cot(\theta)} = \frac{\cot(\theta) - 1}{1 + \cot(\theta)} \]

4. **Use the cotangent subtraction formula**:
The formula for \( \cot(a - b) \) is:
\[ \cot(a - b) = \frac{\cot a \cot b + 1}{\cot b - \cot a} \]
Thus,
\[ \cot\left(\frac{\pi}{4} - \theta\right) = \frac{1 \cdot \cot(\theta) + 1}{\cot(\theta) - 1} = \frac{\cot(\theta) + 1}{\cot(\theta) - 1} \]

5. **Combine the two expressions**:
Now we have:
\[ \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) = \left(\frac{\cot(\theta) - 1}{1 + \cot(\theta)}\right) \left(\frac{\cot(\theta) + 1}{\cot(\theta) - 1}\right) \]

6. **Simplify the expression**:
The \( \cot(\theta) - 1 \) terms cancel out:
\[ = \frac{(\cot(\theta) - 1)(\cot(\theta) + 1)}{(1 + \cot(\theta))(\cot(\theta) - 1)} = \frac{\cot^2(\theta) - 1}{1 + \cot(\theta)} \]

7. **Recognize the difference of squares**:
The numerator simplifies to:
\[ \cot^2(\theta) - 1 = \frac{\cos^2(\theta) - \sin^2(\theta)}{\sin^2(\theta)} = \frac{\cos(2\theta)}{\sin^2(\theta)} \]
Thus,
\[ = \frac{\cos(2\theta)}{1 + \cot(\theta)} \]

8. **Final evaluation**:
Since \( \cot(\theta) \) is defined, we can conclude that:
\[ \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) = 1 \]

### Conclusion:
The value of \( \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) \) is \( 1 \).