The principal value of `cot^(-1)x` lie in

Options

A. `(-pi/2,pi/2)`
B. `[-pi/2,pi/2]`
C. `(0,pi)`
D. `[0,pi]`

Answer

To find the principal value of \( \cot^{-1} x \), we need to determine the range of the inverse cotangent function. Here’s a step-by-step solution:

### Step 1: Understanding the Inverse Function
The function \( \cot^{-1} x \) is the inverse of the cotangent function. The cotangent function, \( \cot \theta \), is defined as the ratio of the adjacent side to the opposite side in a right triangle. The range of \( \cot \theta \) is all real numbers, which means \( \cot^{-1} x \) can take any real number as input.

**Hint:** Recall that the cotangent function is periodic and can take any real value, which is essential for understanding its inverse.

### Step 2: Identifying the Range of \( \cot^{-1} x \)
The principal value of \( \cot^{-1} x \) is defined to be the range of values that \( \cot^{-1} x \) can take. For the inverse cotangent function, the principal values are restricted to a specific interval to ensure that it is a function (i.e., it gives one output for each input).

**Hint:** The range of the inverse cotangent function is typically restricted to avoid ambiguity in values.

### Step 3: Determining the Specific Interval
The principal value of \( \cot^{-1} x \) is defined to lie in the interval \( (0, \pi) \). This means that \( \cot^{-1} x \) can take any value greater than 0 and less than \( \pi \), but it does not include the endpoints.

**Hint:** Remember that the endpoints are not included in the interval, which is a common characteristic of principal values in inverse trigonometric functions.

### Conclusion
Thus, the principal value of \( \cot^{-1} x \) lies in the interval \( (0, \pi) \).

**Final Answer:** The principal value of \( \cot^{-1} x \) lies in \( (0, \pi) \).