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.