If `f (x) = ( cot x )/( 1+ cot x )` and `theta +phi = ( 5pi )/( 4)`, then the value of `f ( theta ). f ( phi )` is

Options

A. `(1)/(2)`
B. `- ( 1)/( 2)`
C. 2
D. `-2`

Answer

To solve the problem, we need to find the value of \( f(\theta) \cdot f(\phi) \) given that \( f(x) = \frac{\cot x}{1 + \cot x} \) and \( \theta + \phi = \frac{5\pi}{4} \).

### Step-by-Step Solution:

1. **Define the function**:
\[ f(x) = \frac{\cot x}{1 + \cot x} \]

2. **Express \( f(\theta) \) and \( f(\phi) \)**:
\[ f(\theta) = \frac{\cot \theta}{1 + \cot \theta} \]
\[ f(\phi) = \frac{\cot \phi}{1 + \cot \phi} \]

3. **Calculate \( f(\theta) \cdot f(\phi) \)**:
\[ f(\theta) \cdot f(\phi) = \left( \frac{\cot \theta}{1 + \cot \theta} \right) \cdot \left( \frac{\cot \phi}{1 + \cot \phi} \right) \]
\[ = \frac{\cot \theta \cdot \cot \phi}{(1 + \cot \theta)(1 + \cot \phi)} \]

4. **Use the cotangent addition formula**:
Since \( \theta + \phi = \frac{5\pi}{4} \), we have:
\[ \cot(\theta + \phi) = \cot\left(\frac{5\pi}{4}\right) = -1 \]
Using the cotangent addition formula:
\[ \cot(\theta + \phi) = \frac{\cot \theta \cdot \cot \phi - 1}{\cot \theta + \cot \phi} \]
Setting this equal to -1 gives:
\[ -1 = \frac{\cot \theta \cdot \cot \phi - 1}{\cot \theta + \cot \phi} \]

5. **Cross-multiply and simplify**:
\[ -(\cot \theta + \cot \phi) = \cot \theta \cdot \cot \phi - 1 \]
Rearranging gives:
\[ \cot \theta \cdot \cot \phi + \cot \theta + \cot \phi - 1 = 0 \]

6. **Let \( A = \cot \theta \) and \( B = \cot \phi \)**:
The equation becomes:
\[ AB + A + B - 1 = 0 \]
Rearranging gives:
\[ AB + A + B = 1 \]

7. **Substituting back**:
We can substitute \( A + B = 1 - AB \) into our expression for \( f(\theta) \cdot f(\phi) \):
\[ f(\theta) \cdot f(\phi) = \frac{AB}{(1 + A)(1 + B)} \]
Expanding the denominator:
\[ (1 + A)(1 + B) = 1 + A + B + AB = 1 + (1 - AB) + AB = 2 \]
Thus:
\[ f(\theta) \cdot f(\phi) = \frac{AB}{2} \]

8. **Using the earlier equation**:
From \( AB + A + B = 1 \), we know:
\[ AB = 1 - (A + B) = 1 - (1 - AB) = 2AB \]
Therefore, \( AB = \frac{1}{2} \).

9. **Final Calculation**:
\[ f(\theta) \cdot f(\phi) = \frac{\frac{1}{2}}{2} = \frac{1}{4} \]

Thus, the value of \( f(\theta) \cdot f(\phi) \) is \( \frac{1}{2} \).