Which of the following is the principal value branch of `cosec^(-1)x` ?

Options

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

Explanation

To determine the principal value branch of \( \csc^{-1} x \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Understanding the Inverse Function**:
We start with the definition of the cosecant function. If \( y = \csc^{-1} x \), then by definition, \( x = \csc y \).

2. **Relating Cosecant to Sine**:
Recall that \( \csc y = \frac{1}{\sin y} \). Therefore, we can write:
\[ x = \frac{1}{\sin y} \implies \sin y = \frac{1}{x} \]

3. **Finding the Range of \( y \)**:
The principal value of \( y \) (which is \( \csc^{-1} x \)) must be defined in a specific range. The sine function is defined for \( y \) in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

4. **Excluding the Undefined Point**:
However, the cosecant function is undefined at \( y = 0 \) because \( \csc 0 \) is undefined. Therefore, we need to exclude this point from our range.

5. **Determining the Final Range**:
Thus, the principal value branch of \( \csc^{-1} x \) is:
\[ y \in \left(-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right) \]

6. **Conclusion**:
The principal value branch of \( \csc^{-1} x \) is:
\[ \text{Principal value branch of } \csc^{-1} x = \left(-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right) \]