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.