Find the domain of definitions of the following function: `f(x)=1/(sqrt(1-cosx))`

Explanation

To find the domain of the function \( f(x) = \frac{1}{\sqrt{1 - \cos x}} \), we need to ensure that the expression under the square root is positive and not equal to zero, since it is in the denominator.

### Step-by-Step Solution:

1. **Identify the condition for the denominator**:
The function \( f(x) \) is defined as long as the denominator \( \sqrt{1 - \cos x} \) is not equal to zero. Therefore, we need:
\[ 1 - \cos x \neq 0 \]
This simplifies to:
\[ \cos x \neq 1 \]

2. **Determine when \( \cos x = 1 \)**:
The cosine function equals 1 at specific points. The general solution for \( \cos x = 1 \) is:
\[ x = 2n\pi \quad \text{where } n \text{ is any integer} \]

3. **Establish the condition for the square root**:
Additionally, since we are dealing with a square root, we need:
\[ 1 - \cos x > 0 \]
This implies:
\[ \cos x < 1 \]

4. **Combine the conditions**:
The condition \( \cos x < 1 \) is satisfied for all \( x \) except where \( \cos x = 1 \). Therefore, the function is defined for all real numbers except at the points where \( x = 2n\pi \).

5. **Write the domain**:
Thus, the domain of the function \( f(x) \) can be expressed as:
\[ \text{Domain of } f(x) = \{ x \in \mathbb{R} \mid x \neq 2n\pi, n \in \mathbb{Z} \} \]

### Final Answer:
The domain of the function \( f(x) = \frac{1}{\sqrt{1 - \cos x}} \) is:
\[ x \in \mathbb{R} \setminus \{ 2n\pi \mid n \in \mathbb{Z} \} \]