To simplify \( \frac{1}{1 - \sin(x)} \), let's try to rewrite it by multiplying both the numerator and denominator by \( 1 + \sin(x) \) to rationalize the denominator. 

\[
\frac{1}{1 - \sin(x)} \cdot \frac{1 + \sin(x)}{1 + \sin(x)} = \frac{1 + \sin(x)}{(1 - \sin(x))(1 + \sin(x))}
\]

The denominator becomes:

\[
(1 - \sin(x))(1 + \sin(x)) = 1 - \sin^2(x)
\]

Using the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \), we can replace \( 1 - \sin^2(x) \) with \( \cos^2(x) \). This gives:

\[
\frac{1 + \sin(x)}{\cos^2(x)}
\]

We can separate this fraction as follows:

\[
\frac{1 + \sin(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)} + \frac{\sin(x)}{\cos^2(x)}
\]

Now, rewrite each term using trigonometric identities:

\[
\frac{1}{\cos^2(x)} = \sec^2(x) \quad \text{and} \quad \frac{\sin(x)}{\cos^2(x)} = \sec(x) \tan(x)
\]

Thus, the simplified form is:

\[
\sec^2(x) + \sec(x) \tan(x)
\]

So,
\[
\frac{1}{1 - \sin(x)} = \sec^2(x) + \sec(x) \tan(x)
\]

Would you like more details, or do you have any questions?

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### Related Questions
1. How would you simplify \( \frac{1}{1 + \sin(x)} \)?
2. What happens if you simplify \( \frac{1}{\cos(x)} \)?
3. Can you simplify \( \frac{1}{1 - \cos(x)} \) similarly?
4. How would you expand \( \sec(x) \cdot \tan(x) \)?
5. How can you use trigonometric identities to simplify \( \frac{\cos(x)}{1 - \sin(x)} \)?

### Tip
To simplify trigonometric expressions, multiplying by a conjugate can help remove terms like \( 1 - \sin(x) \) or \( 1 + \sin(x) \) in the denominator by creating a difference of squares.

Explore the step-by-step solution to simplify the expression 1/(1 - sin(x)) using trigonometric identities and rationalization techniques. This example highlights the Pythagorean identity and is suitable for students in Grades 10-12.

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