If `x lt 0` then `tan^(-1)(1/x)` equals

Options

A. `cot^(-1)x`
B. `-cot^(-1)x`
C. `-pi + cot^(-1)x`
D. `-pi -cot^(-1) x `

Answer

To solve the problem, we need to find the value of \( \tan^{-1}\left(\frac{1}{x}\right) \) given that \( x < 0 \).

### Step-by-Step Solution:

1. **Understanding the Inverse Tangent Function**:
We start with the expression \( \tan^{-1}\left(\frac{1}{x}\right) \). We know that the tangent function is the reciprocal of the cotangent function. Therefore, we can express this as:
\[ \tan^{-1}\left(\frac{1}{x}\right) = \cot^{-1}(x) \]

2. **Using the Property of Cotangent**:
Since \( x < 0 \), we need to consider the properties of the cotangent function. The cotangent function has the following property:
\[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \]
This is useful because \( x \) is negative, so we can express \( \cot^{-1}(x) \) in terms of a positive value.

3. **Substituting the Negative Value**:
We can rewrite our expression using the property mentioned:
\[ \tan^{-1}\left(\frac{1}{x}\right) = \cot^{-1}(x) = \pi - \cot^{-1}(-x) \]
Since \( x < 0 \), we can replace \( x \) with \(-x\):
\[ \tan^{-1}\left(\frac{1}{x}\right) = \pi - \cot^{-1}(-x) \]

4. **Final Expression**:
Therefore, we can conclude that:
\[ \tan^{-1}\left(\frac{1}{x}\right) = \pi - \cot^{-1}(-x) \]
However, since we are looking for the expression in terms of \( x \), we can also express it as:
\[ \tan^{-1}\left(\frac{1}{x}\right) = -\cot^{-1}(x) \]
Thus, the final answer can be represented as:
\[ \tan^{-1}\left(\frac{1}{x}\right) = -\cot^{-1}(x) \]

### Conclusion:
The final result is:
\[ \tan^{-1}\left(\frac{1}{x}\right) = \pi - \cot^{-1}(-x) \]