If `cosec^(-1) ( cosec x) " and " cosec ( cosec^(-1) x) ` are equal functions, then the maximum range of value of x is

Options

A. `[- pi/2 , -1] cup[1, pi/2]`
B. `[- pi/2 , 0 ) cup ( 0, pi/2]`
C. `( - infty, -1] cup [1, infty)`
D. `[-1, 0) cup (0,1]`

Answer

To solve the problem, we need to analyze the functions \( \csc^{-1}(\csc x) \) and \( \csc(\csc^{-1} x) \) and determine the conditions under which they are equal.

### Step 1: Understanding the Functions

1. **Function 1: \( \csc^{-1}(\csc x) \)**
- The function \( \csc^{-1}(y) \) is defined for \( y \leq -1 \) or \( y \geq 1 \).
- For \( \csc^{-1}(\csc x) \) to be defined, \( x \) must be in the range where \( \csc x \) is valid, which is \( x \in (-\frac{\pi}{2}, \frac{\pi}{2}) \) excluding \( x = 0 \).

2. **Function 2: \( \csc(\csc^{-1} x) \)**
- The function \( \csc^{-1}(x) \) is defined for \( x \leq -1 \) or \( x \geq 1 \).
- Therefore, \( \csc(\csc^{-1} x) = x \) holds true for \( x \leq -1 \) or \( x \geq 1 \).

### Step 2: Setting Up the Equality

We need to find the values of \( x \) for which:
\[ \csc^{-1}(\csc x) = \csc(\csc^{-1} x) \]

### Step 3: Finding the Range of \( x \)

1. **Range of \( \csc^{-1}(\csc x) \)**:
- From the analysis, \( \csc^{-1}(\csc x) \) is defined for \( x \in (-\frac{\pi}{2}, \frac{\pi}{2}) \setminus \{0\} \).

2. **Range of \( \csc(\csc^{-1} x) \)**:
- This function is defined for \( x \leq -1 \) or \( x \geq 1 \).

### Step 4: Finding the Intersection of Ranges

To find the maximum range of \( x \) such that both functions are equal, we need to find the intersection of the two ranges:

- The range from \( \csc^{-1}(\csc x) \) is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \setminus \{0\} \).
- The range from \( \csc(\csc^{-1} x) \) is \( (-\infty, -1] \cup [1, \infty) \).

### Step 5: Conclusion

The intersection of these ranges gives us:
- From \( (-\frac{\pi}{2}, \frac{\pi}{2}) \setminus \{0\} \), the values that satisfy \( x \leq -1 \) or \( x \geq 1 \) are:
- \( x \in [1, \frac{\pi}{2}) \)

Thus, the maximum range of value of \( x \) is:
\[ \text{Maximum range of } x = [1, \frac{\pi}{2}) \]

### Final Answer

The maximum range of value of \( x \) is \( [1, \frac{\pi}{2}) \).

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