It seems like you're looking to simplify the expression:

$\frac{\cos(\theta)}{1 - \sin(\theta)}$

This expression is quite general, so we can look at potential simplifications or manipulations depending on what you're aiming for. Let's break it down:

Possible manipulations:

1. Multiplying the numerator and denominator by the conjugate of the denominator: This is a common method for simplifying expressions involving trigonometric functions like this. The conjugate of $1 - \sin(\theta)$ is $1 + \sin(\theta)$. So, we multiply both the numerator and denominator by $1 + \sin(\theta)$:

$\frac{\cos(\theta)}{1 - \sin(\theta)} \times \frac{1 + \sin(\theta)}{1 + \sin(\theta)} = \frac{\cos(\theta)(1 + \sin(\theta))}{(1 - \sin(\theta))(1 + \sin(\theta))}$

2. Simplifying the denominator: The denominator is now a difference of squares:

$(1 - \sin(\theta))(1 + \sin(\theta)) = 1^2 - \sin^2(\theta) = \cos^2(\theta)$

So the expression becomes:

$\frac{\cos(\theta)(1 + \sin(\theta))}{\cos^2(\theta)}$

3. Simplifying further: You can cancel one factor of $\cos(\theta)$ from the numerator and denominator (assuming $\cos(\theta) \neq 0$):

$\frac{1 + \sin(\theta)}{\cos(\theta)}$

Final result:

$\frac{1 + \sin(\theta)}{\cos(\theta)}$

This expression can also be rewritten as:

$\sec(\theta) + \tan(\theta)$

This is the simplified form.

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Would you like to explore more about trigonometric identities or the steps involved here? Let me know if you'd like further explanations on any part!

Here are some related questions for further exploration:

1. How can we simplify trigonometric expressions involving both sine and cosine?
2. What is the significance of using conjugates in simplifying expressions?
3. How do we prove trigonometric identities using algebraic manipulation?
4. What is the relationship between $\sec(\theta)$, $\tan(\theta)$, and the unit circle?
5. How can we use the Pythagorean identity to simplify more complex trigonometric expressions?

Tip: Whenever simplifying trigonometric expressions, consider multiplying by conjugates when you see terms like $1 - \sin(\theta)$ or $1 + \sin(\theta)$, as it can often lead to cleaner results!