The value of `cot^(-1)(-x) x in R` in terms of `cot^(-1)x` is `"…….."`

Explanation

To find the value of \( \cot^{-1}(-x) \) in terms of \( \cot^{-1}(x) \), we can use the properties of the inverse cotangent function.

### Step-by-step Solution:

1. **Understanding the Inverse Cotangent Function**:
The inverse cotangent function, \( \cot^{-1}(x) \), gives us an angle \( \theta \) such that \( \cot(\theta) = x \). The range of \( \cot^{-1}(x) \) is \( (0, \pi) \).

2. **Using the Property of Cotangent**:
We know that:
\[ \cot(-\theta) = -\cot(\theta) \]
This means that if \( \theta = \cot^{-1}(x) \), then:
\[ \cot(-\theta) = -x \]
Hence, we can express \( \cot^{-1}(-x) \) in terms of \( \theta \).

3. **Expressing \( \cot^{-1}(-x) \)**:
If \( \theta = \cot^{-1}(x) \), then:
\[ \cot^{-1}(-x) = -\theta \]
However, since we want the angle to be within the range \( (0, \pi) \), we can express this as:
\[ \cot^{-1}(-x) = \pi - \theta \]

4. **Substituting Back**:
Substituting \( \theta \) back, we have:
\[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \]

### Final Answer:
Thus, the value of \( \cot^{-1}(-x) \) in terms of \( \cot^{-1}(x) \) is:
\[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \]