To determine the domain of the function \( f(x) = \frac{\text{cost}}{1 - \sin(x)} \), we need to find all values of \( x \) for which the function is defined.

The function is undefined wherever the denominator is zero. Therefore, we need to 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 \( x = \frac{\pi}{2} + 2n\pi \). The domain of the function is all real numbers except these values.

**Domain:** \( x \in \mathbb{R} \) except \( x \neq \frac{\pi}{2} + 2n\pi \), where \( n \) is any integer.

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

Here are some related questions:
1. What is the range of \( f(x) = \frac{\text{cost}}{1 - \sin(x)} \)?
2. How do you find the domain of a more complex trigonometric function?
3. What are the general solutions to \( \sin(x) = 0 \)?
4. What is the significance of the period of the sine function in determining the domain?
5. How does the sine function behave near \( \frac{\pi}{2} \)?

**Tip:** For trigonometric functions, always check where the denominator might be zero to determine the domain.

Learn how to determine the domain of the function f(x) = cosθ / (1 - sin(x)) with detailed step-by-step explanations. Explore where the function is defined and where it is undefined due to the characteristics of trigonometric functions. Suitable for advanced high school students studying trigonometry.

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