`cot((pi)/(4)+theta)cot((pi)/(4)-theta)=?`

Options

A. `-1`
B. 1
C. 0
D. `oo`

Answer

To solve the expression \( \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) \), we can use the cotangent addition and subtraction formulas. Let's break it down step by step.

### Step-by-Step Solution:

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

2. **Use the cotangent addition formula**:
Recall the cotangent addition formula:
\[ \cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} \]
For \( A = \frac{\pi}{4} \) and \( B = \theta \):
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{\cot\left(\frac{\pi}{4}\right) \cot(\theta) - 1}{\cot\left(\frac{\pi}{4}\right) + \cot(\theta)} \]
Since \( \cot\left(\frac{\pi}{4}\right) = 1 \), we have:
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{1 \cdot \cot(\theta) - 1}{1 + \cot(\theta)} = \frac{\cot(\theta) - 1}{1 + \cot(\theta)} \]

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

4. **Multiply the two cotangent expressions**:
Now we multiply the two results:
\[ \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) = \left(\frac{\cot(\theta) - 1}{1 + \cot(\theta)}\right) \left(\frac{-\cot(\theta) - 1}{1 - \cot(\theta)}\right) \]

5. **Simplify the expression**:
We can simplify this expression further. However, a more straightforward approach is to use the identity:
\[ \cot(A + B) \cot(A - B) = 1 \]
when \( A = \frac{\pi}{4} \) and \( B = \theta \). Thus:
\[ \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) = 1 \]

### Final Answer:
Thus, the final result is:
\[ \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) = 1 \]