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We are tasked with proving the trigonometric identity:

\[
\frac{\cos(x)}{1 - \sin(x)} = \sec(x) + \tan(x)
\]

### Proof:

1. **Start with the right-hand side** (RHS):
   \[
   \sec(x) + \tan(x)
   \]
   Recall that:
   \[
   \sec(x) = \frac{1}{\cos(x)} \quad \text{and} \quad \tan(x) = \frac{\sin(x)}{\cos(x)}
   \]
   Therefore:
   \[
   \sec(x) + \tan(x) = \frac{1}{\cos(x)} + \frac{\sin(x)}{\cos(x)}
   \]
   Combine the terms over a common denominator:
   \[
   \sec(x) + \tan(x) = \frac{1 + \sin(x)}{\cos(x)}
   \]

2. **Simplify the left-hand side** (LHS):
   The LHS is:
   \[
   \frac{\cos(x)}{1 - \sin(x)}
   \]
   
3. **Multiply the numerator and denominator by** \(1 + \sin(x)\):
   This step is designed to simplify the denominator using a difference of squares. We have:
   \[
   \frac{\cos(x)}{1 - \sin(x)} \cdot \frac{1 + \sin(x)}{1 + \sin(x)} = \frac{\cos(x)(1 + \sin(x))}{(1 - \sin(x))(1 + \sin(x))}
   \]
   Simplifying the denominator:
   \[
   (1 - \sin(x))(1 + \sin(x)) = 1 - \sin^2(x)
   \]
   And using the Pythagorean identity \(1 - \sin^2(x) = \cos^2(x)\), we get:
   \[
   \frac{\cos(x)(1 + \sin(x))}{\cos^2(x)}
   \]

4. **Simplify the expression**:
   Cancel out one factor of \(\cos(x)\):
   \[
   \frac{1 + \sin(x)}{\cos(x)}
   \]

This matches the RHS, so the identity is proven:

\[
\frac{\cos(x)}{1 - \sin(x)} = \sec(x) + \tan(x)
\]

### Conclusion:
The trigonometric identity is successfully verified.

Do you have any questions or would you like a detailed explanation of any step? Here are 5 related questions for further exploration:

1. Can you prove the identity \(\frac{1 - \sin(x)}{\cos(x)} = \sec(x) - \tan(x)\)?
2. What is the derivation of the Pythagorean identity \(1 - \sin^2(x) = \cos^2(x)\)?
3. How can the secant and tangent functions be expressed in terms of sine and cosine?
4. What are some practical applications of trigonometric identities in physics or engineering?
5. How would the identity change if we used \(\cot(x)\) and \(\csc(x)\) instead of \(\tan(x)\) and \(\sec(x)\)?

**Tip**: Always look for opportunities to multiply by a conjugate when simplifying complex trigonometric expressions—it often reveals useful identities like the difference of squares.

Prove the identity \( \frac{\cos(x)}{1 - \sin(x)} = \sec(x) + \tan(x) \)

Step-by-step solution to proving the trigonometric identity \( \frac{\cos(x)}{1 - \sin(x)} = \sec(x) + \tan(x) \). This problem involves key concepts in trigonometric identities and algebraic manipulation, suitable for students in Grades 10-12.

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