If ` (-pi)/(2) < theta < (pi)/(2) " and " theta ne pm (pi)/(4),` then the value of ` cot ((pi)/(4) + theta) cot ((pi)/(4) - theta)` is

Options

A. 0
B. `-1`
C. 1
D. `-2`

Answer

To solve the problem, we need to evaluate the expression \( \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) \).

### Step 1: Use the cotangent addition and subtraction formulas
We can use the formulas for cotangent of sums and differences:
\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]
\[ \cot(a - b) = \frac{\cot a \cot b + 1}{\cot a - \cot b} \]
In our case, let \( a = \frac{\pi}{4} \) and \( b = \theta \).

### Step 2: Calculate \( \cot\left(\frac{\pi}{4} + \theta\right) \)
Using the formula for \( \cot(a + b) \):
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{\cot\left(\frac{\pi}{4}\right) \cot(\theta) - 1}{\cot\left(\frac{\pi}{4}\right) + \cot(\theta)} \]
Since \( \cot\left(\frac{\pi}{4}\right) = 1 \):
\[ \cot\left(\frac{\pi}{4} + \theta\right) = \frac{1 \cdot \cot(\theta) - 1}{1 + \cot(\theta)} = \frac{\cot(\theta) - 1}{1 + \cot(\theta)} \]

### Step 3: Calculate \( \cot\left(\frac{\pi}{4} - \theta\right) \)
Using the formula for \( \cot(a - b) \):
\[ \cot\left(\frac{\pi}{4} - \theta\right) = \frac{\cot\left(\frac{\pi}{4}\right) \cot(\theta) + 1}{\cot\left(\frac{\pi}{4}\right) - \cot(\theta)} \]
Again, since \( \cot\left(\frac{\pi}{4}\right) = 1 \):
\[ \cot\left(\frac{\pi}{4} - \theta\right) = \frac{1 \cdot \cot(\theta) + 1}{1 - \cot(\theta)} = \frac{\cot(\theta) + 1}{1 - \cot(\theta)} \]

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

### Step 5: Simplify the expression
Using the difference of squares:
\[ = \frac{(\cot(\theta) - 1)(\cot(\theta) + 1)}{(1 + \cot(\theta))(1 - \cot(\theta))} = \frac{\cot^2(\theta) - 1}{1 - \cot^2(\theta)} \]
Recognizing that \( \cot^2(\theta) - 1 = -\sin^2(\theta) \) and \( 1 - \cot^2(\theta) = \sin^2(\theta) \):
\[ = -1 \]

### Final Answer
Thus, the value of \( \cot\left(\frac{\pi}{4} + \theta\right) \cot\left(\frac{\pi}{4} - \theta\right) \) is \( -1 \).