The domain of the function `f(x)=sqrt(log((1)/(|sinx|)))`

Options

A. `R-{-pi,pi}`
B. `R-{n pi|n in Z}`
C. `R-{2n pi|n in z}`
D. `(-oo,oo)`

Explanation

To find the domain of the function \( f(x) = \sqrt{\log\left(\frac{1}{|\sin x|}\right)} \), we need to ensure that the expression inside the square root is non-negative and that the logarithm is defined.

### Step 1: Define the conditions for the logarithm
The logarithm function \( \log(y) \) is defined for \( y > 0 \). Therefore, we need:
\[ \frac{1}{|\sin x|} > 0 \]
This condition is satisfied as long as \( |\sin x| \neq 0 \).

### Step 2: Determine when \( |\sin x| \) is zero
The modulus function \( |\sin x| \) is zero when \( \sin x = 0 \). This occurs at:
\[ x = n\pi \quad \text{where } n \in \mathbb{Z} \]
Thus, we must exclude these points from the domain.

### Step 3: Condition for the logarithm to be positive
Next, we need the logarithm itself to be positive:
\[ \log\left(\frac{1}{|\sin x|}\right) > 0 \]
This implies:
\[ \frac{1}{|\sin x|} > 1 \]
Taking the reciprocal gives:
\[ |\sin x| < 1 \]
Since \( |\sin x| \) is always between 0 and 1 for all \( x \) (except at points where \( \sin x = 0 \)), this condition is satisfied for all \( x \) except where \( \sin x = 0 \).

### Step 4: Combine the conditions
Thus, the domain of \( f(x) \) is all real numbers except where \( \sin x = 0 \):
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \{ n\pi \mid n \in \mathbb{Z} \} \]

### Final Answer
The domain of the function \( f(x) = \sqrt{\log\left(\frac{1}{|\sin x|}\right)} \) is:
\[ \mathbb{R} - \{ n\pi \mid n \in \mathbb{Z} \} \]