We can simplify the expression \(\frac{\sec(\theta) - \cos(\theta)}{\sin(\theta)}\) as follows:

1. **Recall the identity for secant**:
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)}
   \]

2. **Substitute this into the original expression**:
   \[
   \frac{\frac{1}{\cos(\theta)} - \cos(\theta)}{\sin(\theta)}
   \]

3. **Simplify the numerator**:
   \[
   \frac{1 - \cos^2(\theta)}{\cos(\theta)}
   \]

4. **Use the Pythagorean identity**: 
   \[
   1 - \cos^2(\theta) = \sin^2(\theta)
   \]
   So, the expression becomes:
   \[
   \frac{\sin^2(\theta)}{\cos(\theta)}
   \]

5. **Divide by \(\sin(\theta)\) from the denominator**:
   \[
   \frac{\sin^2(\theta)}{\cos(\theta) \cdot \sin(\theta)} = \frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta)
   \]

Thus, the simplified expression is:
\[
\tan(\theta)
\]

Would you like more details on any of the steps?

Here are some related questions:
1. What is the identity for secant and how is it derived?
2. How does the Pythagorean identity help in simplifying trigonometric expressions?
3. What other expressions simplify to \(\tan(\theta)\)?
4. How can we apply these identities to solve real-world problems?
5. How does this type of simplification help in calculus?

**Tip**: Memorizing common trigonometric identities makes solving problems involving simplifications much faster.

Learn how to simplify the trigonometric expression (sec(θ) - cos(θ)) / sin(θ) to tan(θ). This solution uses key trigonometric identities, such as the secant identity and the Pythagorean identity, making it suitable for high school students in Grades 10-12.

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