Find domain for `f(x)=[sinx] cos(pi/([x-1]))`.

Answer

To find the domain of the function \( f(x) = [\sin x] \cos\left(\frac{\pi}{x-1}\right) \), we need to analyze the components of the function.

### Step 1: Identify the components of the function
The function consists of two parts:
1. The greatest integer function \( [\sin x] \)
2. The cosine function \( \cos\left(\frac{\pi}{x-1}\right) \)

### Step 2: Determine the domain of \( [\sin x] \)
The greatest integer function \( [\sin x] \) is defined for all real numbers \( x \). Therefore, there are no restrictions on \( x \) from this part.

### Step 3: Determine the domain of \( \cos\left(\frac{\pi}{x-1}\right) \)
The cosine function is defined for all real numbers, but we need to consider the argument \( \frac{\pi}{x-1} \). The expression \( \frac{\pi}{x-1} \) is undefined when the denominator is zero. This occurs when:
\[ x - 1 = 0 \implies x = 1 \]
Thus, \( x \) cannot be equal to 1.

### Step 4: Analyze the behavior around \( x = 1 \)
The function \( \cos\left(\frac{\pi}{x-1}\right) \) will approach infinity as \( x \) approaches 1 from either side, which indicates a discontinuity at this point. Therefore, we need to exclude \( x = 1 \) from the domain.

### Step 5: Combine the restrictions
Since \( [\sin x] \) does not impose any restrictions and \( \cos\left(\frac{\pi}{x-1}\right) \) is undefined at \( x = 1 \), the domain of the function is all real numbers except for 1.

### Conclusion
The domain of the function \( f(x) = [\sin x] \cos\left(\frac{\pi}{x-1}\right) \) is:
\[ \text{Domain} = \mathbb{R} \setminus \{1\} \]