To determine the domain of the function \( f(x) = \frac{\cos(x)}{1 - \sin(x)} \), we need to ensure that the denominator is not equal to zero, as division by zero is undefined.

The function is undefined wherever the denominator is zero. Therefore, we solve the equation:

\[
1 - \sin(x) = 0
\]

This simplifies to:

\[
\sin(x) = 1
\]

The sine function equals 1 at specific points:

\[
x = \frac{\pi}{2} + 2n\pi \quad \text{where } n \text{ is any integer}
\]

Thus, the function \( f(x) \) is undefined at \( x = \frac{\pi}{2} + 2n\pi \). 

**Domain:** The domain of the function is all real numbers except \( x = \frac{\pi}{2} + 2n\pi \), where \( n \) is any integer.

Would you like more details on this or have any questions?

Here are some related questions:
1. How does the period of the sine and cosine functions affect the domain of their ratios?
2. What is the range of \( f(x) = \frac{\cos(x)}{1 - \sin(x)} \)?
3. How do you find the domain of a function involving both sine and cosine?
4. What are the general solutions to \( \sin(x) = 1 \)?
5. How does the behavior of \( \cos(x) \) near \( \frac{\pi}{2} \) influence the function?

**Tip:** When analyzing trigonometric functions, always consider the periodicity and key points where the function could become undefined.

Explore the domain of the function f(x) = cos(x)/(1 - sin(x)). Learn why the function is undefined at specific points and how to determine its domain using trigonometric identities and equations. Suitable for advanced high school students and anyone interested in trigonometry.

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