The domain of the function `f (x) = sqrt"" [log (1//| sin x |)]` is

Options

A. ` R - {2 n pi , n in Z}`
B. `R-{ n pi , n in Z}`
C. ` R-{- pi ,pi}`
D. None of these

Answer

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: Determine the conditions for the logarithm
The logarithmic function \( \log(z) \) is defined only for \( z > 0 \). In our case, we have:
\[ \frac{1}{|\sin x|} > 0 \]
This condition is satisfied as long as \( |\sin x| \neq 0 \). Therefore, we need:
\[ |\sin x| \neq 0 \]
This means that \( \sin x \) cannot be equal to 0.

### Step 2: Identify where \( \sin x = 0 \)
The sine function is equal to zero at integer multiples of \( \pi \):
\[ x \neq n\pi \quad \text{where } n \in \mathbb{Z} \]

### Step 3: Determine the conditions for the square root
Next, we need to ensure that the expression inside the square root is non-negative:
\[ \log\left(\frac{1}{|\sin x|}\right) \geq 0 \]
This implies:
\[ \frac{1}{|\sin x|} \geq 1 \]
Taking the reciprocal (and noting that since \( |\sin x| > 0 \), the inequality direction remains the same):
\[ |\sin x| \leq 1 \]
This condition is always satisfied since the sine function oscillates between -1 and 1. However, we must also ensure that \( |\sin x| \neq 0 \).

### Step 4: Combine the conditions
From the above steps, we have:
1. \( |\sin x| \neq 0 \) (i.e., \( x \neq n\pi \))
2. \( |\sin x| \leq 1 \) (which is always true)

Thus, the only restriction on \( x \) comes from the first condition.

### Conclusion
The domain of the function \( f(x) \) is:
\[ \text{Domain: } x \in \mathbb{R} \setminus \{ n\pi \mid n \in \mathbb{Z} \} \]