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 and tangent.

### Step-by-step Solution:

1. **Use the Cotangent Addition Formula**:
\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]
Here, let \( a = \frac{\pi}{4} \) and \( b = \theta \).
Thus, we have:
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{\cot\left(\frac{\pi}{4}\right) \cot(\theta) - 1}{\cot\left(\frac{\pi}{4}\right) + \cot(\theta)} \]

2. **Substituting \( \cot\left(\frac{\pi}{4}\right) \)**:
We know that \( \cot\left(\frac{\pi}{4}\right) = 1 \). So,
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{1 \cdot \cot(\theta) - 1}{1 + \cot(\theta)} = \frac{\cot(\theta) - 1}{1 + \cot(\theta)} \]

3. **Use the Cotangent Subtraction Formula**:
Similarly, for \( \cot\left(\frac{\pi}{4} - \theta\right) \):
\[ \cot\left(\frac{\pi}{4} - \theta\right) = \frac{\cot\left(\frac{\pi}{4}\right) \cot(\theta) + 1}{\cot\left(\frac{\pi}{4}\right) - \cot(\theta)} = \frac{1 \cdot \cot(\theta) + 1}{1 - \cot(\theta)} = \frac{\cot(\theta) + 1}{1 - \cot(\theta)} \]

4. **Multiply the Two Cotangent Values**:
Now, we need to find:
\[ \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) = \left(\frac{\cot(\theta) - 1}{1 + \cot(\theta)}\right) \left(\frac{\cot(\theta) + 1}{1 - \cot(\theta)}\right) \]

5. **Simplify the Expression**:
Using the difference of squares:
\[ = \frac{(\cot(\theta) - 1)(\cot(\theta) + 1)}{(1 + \cot(\theta))(1 - \cot(\theta))} = \frac{\cot^2(\theta) - 1}{1 - \cot^2(\theta)} \]
Since \( \cot^2(\theta) - 1 = - (1 - \cot^2(\theta)) \), we can simplify this to:
\[ = -1 \]

6. **Final Result**:
Thus, the value of \( \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) \) is:
\[ \boxed{1} \]