`cosec^(-1)(cos x)` is real if

Options

A. `x in [-1,1]`
B. `x in R`
C. x is an odd multiple of `(pi)/(2)`
D. None of these

Answer

To determine when \( \csc^{-1}(\cos x) \) is real, we need to analyze the conditions under which the expression is defined. The cosecant inverse function, \( \csc^{-1}(y) \), is defined for \( y \) such that \( |y| \geq 1 \). Therefore, we need to find when \( |\cos x| \geq 1 \).

### Step-by-Step Solution:

1. **Understanding the Cosecant Function**:
The cosecant function is defined as:
\[ \csc x = \frac{1}{\sin x} \]
Thus, \( \csc^{-1}(y) \) is defined for \( |y| \geq 1 \).

2. **Analyzing \( \cos x \)**:
The cosine function, \( \cos x \), has a range of values between -1 and 1:
\[ -1 \leq \cos x \leq 1 \]

3. **Setting the Condition**:
For \( \csc^{-1}(\cos x) \) to be real, we require:
\[ |\cos x| \geq 1 \]
This means \( \cos x \) must be either \( 1 \) or \( -1 \).

4. **Finding Values of \( x \)**:
The cosine function equals \( 1 \) at:
\[ x = 2n\pi \quad \text{(where \( n \) is any integer)} \]
The cosine function equals \( -1 \) at:
\[ x = (2n + 1)\pi \quad \text{(where \( n \) is any integer)} \]

5. **Conclusion**:
Therefore, \( \csc^{-1}(\cos x) \) is real if:
\[ x = n\pi \quad \text{(where \( n \) is any integer)} \]

### Final Answer:
The function \( \csc^{-1}(\cos x) \) is real if \( x \) is an integer multiple of \( \pi \).

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