1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Recall formulas and identities:** - $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$ - Pythagorean identity: $$\sin^2(\theta) + \cos^2(\theta) = 1$$ 3. **Rewrite the expression:** $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \frac{\sin(\theta)}{\cos(\theta)}$$ 4. **Find a common denominator:** The common denominator is $$\cos(\theta)(1 - \sin(\theta))$$. 5. **Rewrite each term with the common denominator:** $$\frac{\cos^2(\theta)}{\cos(\theta)(1 - \sin(\theta))} - \frac{\sin(\theta)(1 - \sin(\theta))}{\cos(\theta)(1 - \sin(\theta))}$$ 6. **Combine the fractions:** $$\frac{\cos^2(\theta) - \sin(\theta)(1 - \sin(\theta))}{\cos(\theta)(1 - \sin(\theta))}$$ 7. **Expand the numerator:** $$\cos^2(\theta) - \sin(\theta) + \sin^2(\theta)$$ 8. **Use the Pythagorean identity:** Since $$\cos^2(\theta) + \sin^2(\theta) = 1$$, numerator becomes: $$1 - \sin(\theta)$$ 9. **Simplify the fraction:** $$\frac{1 - \sin(\theta)}{\cos(\theta)(1 - \sin(\theta))}$$ 10. **Cancel common factors:** $$1 - \sin(\theta)$$ cancels out, leaving: $$\frac{1}{\cos(\theta)}$$ 11. **Final answer:** $$\boxed{\sec(\theta)}$$ This means the original expression simplifies to $$\sec(\theta)$$, which is the reciprocal of cosine.