Question

Complete the following two-column proof. Off to the side of each step write the trig identity or simplification process you are using for each step. You will be graded for showing the full process and getting the correct answer.

Column 1:
tan(theta) / sin(theta)


Column 2:
sec(theta)

Answer 1

To complete your two-column proof showing that \(\frac{\tan(\theta)}{\sin(\theta)} = \sec(\theta)\), we'll present the steps in the left column and the corresponding trig identities or simplifications in the right column.

### Two-Column Proof

| Column 1 | Column 2 |
|----------------------------------------|------------------------------------------------------------|
| 1. \(\frac{\tan(\theta)}{\sin(\theta)}\) | Given |
| 2. \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) | Definition of tangent |
| 3. Substitute: \(\frac{\frac{\sin(\theta)}{\cos(\theta)}}{\sin(\theta)}\) | Substitution from step 2 |
| 4. \(\frac{\sin(\theta)}{\cos(\theta)} \div \sin(\theta)\) | Simplification |
| 5. \(\frac{\sin(\theta)}{\sin(\theta) \cdot \cos(\theta)}\) | Rewrite as division of fractions |
| 6. \( \frac{1}{\cos(\theta)} \) | Cancel \(\sin(\theta)\) (provided \(\sin(\theta) \neq 0\)) |
| 7. \(\sec(\theta)\) | Definition of secant (i.e., \(\sec(\theta) = \frac{1}{\cos(\theta)}\)) |

### Final Conclusion
Thus, we have shown that \(\frac{\tan(\theta)}{\sin(\theta)} = \sec(\theta)\).

### Key Trig Identities Used
1. Definition of tangent: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
2. Definition of secant: \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
3. Simplifying fractions by canceling similar terms.

This completes the proof!