1. **State the problem:** Simplify the expression \( \frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta) \). 2. **Recall formulas and identities:** - \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) - Important Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \) 3. **Rewrite the expression using the identity for \( \tan(\theta) \):** $$ \frac{\cos(\theta)}{1 - \sin(\theta)} - \frac{\sin(\theta)}{\cos(\theta)} $$ 4. **Find a common denominator to combine the terms:** The common denominator is \( (1 - \sin(\theta)) \cos(\theta) \). 5. **Rewrite each term with the common denominator:** $$ \frac{\cos^2(\theta)}{(1 - \sin(\theta)) \cos(\theta)} - \frac{\sin(\theta)(1 - \sin(\theta))}{(1 - \sin(\theta)) \cos(\theta)} $$ 6. **Combine the numerators:** $$ \frac{\cos^2(\theta) - \sin(\theta) + \sin^2(\theta)}{(1 - \sin(\theta)) \cos(\theta)} $$ 7. **Use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) to simplify the numerator:** $$ \frac{1 - \sin(\theta)}{(1 - \sin(\theta)) \cos(\theta)} $$ 8. **Cancel the common factor \( 1 - \sin(\theta) \) in numerator and denominator:** $$ \frac{1}{\cos(\theta)} $$ 9. **Final simplified expression:** $\sec(\theta)$ This means the original expression simplifies to the secant of \( \theta \).