The value of `cot(180+theta)cot(90-theta)` is:

Options

A. 1
B. -1
C. 0
D. none of these

Answer

To solve the problem of finding the value of \( \cot(180^\circ + \theta) \cdot \cot(90^\circ - \theta) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the cotangent of \( 180^\circ + \theta \)**:
\[ \cot(180^\circ + \theta) = -\cot(\theta) \]
This is because the cotangent function is negative in the third quadrant.

2. **Identify the cotangent of \( 90^\circ - \theta \)**:
\[ \cot(90^\circ - \theta) = \tan(\theta) \]
This is due to the co-function identity, where \( \cot(90^\circ - x) = \tan(x) \).

3. **Combine the two results**:
\[ \cot(180^\circ + \theta) \cdot \cot(90^\circ - \theta) = (-\cot(\theta)) \cdot \tan(\theta) \]

4. **Use the identity \( \cot(\theta) = \frac{1}{\tan(\theta)} \)**:
\[ (-\cot(\theta)) \cdot \tan(\theta) = -\left(\frac{1}{\tan(\theta)}\right) \cdot \tan(\theta) \]

5. **Simplify the expression**:
\[ -\left(\frac{1}{\tan(\theta)} \cdot \tan(\theta)\right) = -1 \]

### Final Result:
Thus, the value of \( \cot(180^\circ + \theta) \cdot \cot(90^\circ - \theta) \) is:
\[ \boxed{-1} \]