Domain of the function `f(x)=(1)/([sinx-1])` (where [.] denotes the greatest integer function) is

Options

A. `R~{2npi+(pi)/(2), n in l}`
B. `R~{npi+(pi)/(2), n in l)}`
C. `[2n pi, 2npi+(pi)/(2)]`
D. none of these

Answer

To find the domain of the function \( f(x) = \frac{1}{[\sin x - 1]} \), where \([\cdot]\) denotes the greatest integer function, we need to determine when the denominator is defined and non-zero.

### Step-by-Step Solution:

1. **Identify when the denominator is zero**:
The function \( f(x) \) is undefined when the denominator \( [\sin x - 1] = 0 \). Therefore, we need to find when \( \sin x - 1 = 0 \).

\[ \sin x - 1 = 0 \implies \sin x = 1 \]

2. **Find the values of \( x \) where \( \sin x = 1 \)**:
The sine function equals 1 at specific points. The general solution for \( \sin x = 1 \) is:

\[ x = \frac{\pi}{2} + 2n\pi \quad \text{for } n \in \mathbb{Z} \]

This means that \( f(x) \) is undefined at these points.

3. **Determine the domain**:
Since \( f(x) \) is defined for all real numbers except where \( \sin x = 1 \), the domain of \( f(x) \) can be expressed as:

\[ \text{Domain of } f(x) = \mathbb{R} \setminus \left\{ \frac{\pi}{2} + 2n\pi \mid n \in \mathbb{Z} \right\} \]

### Final Answer:
The domain of the function \( f(x) = \frac{1}{[\sin x - 1]} \) is:

\[ \mathbb{R} - \left\{ 2n\pi + \frac{\pi}{2} \mid n \in \mathbb{Z} \right\} \]