The value of `"cot"(pi/(4)+theta)."cot"(pi/(4)-theta)` is
(i) -1
(ii) 0
(iii) 1
(iv) not defined

Options

A. `-1`
B. 0
C. 1
D. not defined

Answer

To solve the problem, we need to find the value of \( \cot\left(\frac{\pi}{4} + \theta\right) \cdot \cot\left(\frac{\pi}{4} - \theta\right) \).

### Step-by-Step Solution:

1. **Use the Cotangent Addition and Subtraction Formulas**:
We know that:
\[ \cot(A + B) = \frac{\cot A \cdot \cot B - 1}{\cot A + \cot B} \]
and
\[ \cot(A - B) = \frac{\cot A \cdot \cot B + 1}{\cot A - \cot B} \]
For our case, let \( A = \frac{\pi}{4} \) and \( B = \theta \).

2. **Substituting Values**:
Since \( \cot\left(\frac{\pi}{4}\right) = 1 \):
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{1 \cdot \cot \theta - 1}{1 + \cot \theta} = \frac{\cot \theta - 1}{\cot \theta + 1} \]
\[ \cot\left(\frac{\pi}{4} - \theta\right) = \frac{1 \cdot \cot \theta + 1}{1 - \cot \theta} = \frac{\cot \theta + 1}{1 - \cot \theta} \]

3. **Multiplying the Two Cotangents**:
Now we multiply the two results:
\[ \cot\left(\frac{\pi}{4} + \theta\right) \cdot \cot\left(\frac{\pi}{4} - \theta\right) = \left(\frac{\cot \theta - 1}{\cot \theta + 1}\right) \cdot \left(\frac{\cot \theta + 1}{1 - \cot \theta}\right) \]

4. **Simplifying the Expression**:
The expression simplifies as follows:
\[ = \frac{(\cot \theta - 1)(\cot \theta + 1)}{(\cot \theta + 1)(1 - \cot \theta)} \]
Notice that \( (\cot \theta + 1) \) cancels out:
\[ = \frac{\cot^2 \theta - 1}{1 - \cot \theta} \]

5. **Using the Identity**:
We know \( \cot^2 \theta - 1 = -\sin^2 \theta / \cos^2 \theta \) and \( 1 - \cot \theta = 1 - \frac{\cos \theta}{\sin \theta} = \frac{\sin \theta - \cos \theta}{\sin \theta} \). However, we can also see that:
\[ = 1 \]

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