1. **State the problem:** Simplify and verify the identity $$\frac{\sin \theta}{1 - \cos \theta} - \cot \theta = \csc \theta.$$ 2. **Recall formulas and identities:** - $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$ - $$\csc \theta = \frac{1}{\sin \theta}$$ - Use the Pythagorean identity: $$1 - \cos^2 \theta = \sin^2 \theta$$ 3. **Rewrite the left side:** $$\frac{\sin \theta}{1 - \cos \theta} - \cot \theta = \frac{\sin \theta}{1 - \cos \theta} - \frac{\cos \theta}{\sin \theta}.$$ 4. **Find a common denominator:** The common denominator is $$\sin \theta (1 - \cos \theta).$$ Rewrite each term: $$\frac{\sin^2 \theta}{\sin \theta (1 - \cos \theta)} - \frac{\cos \theta (1 - \cos \theta)}{\sin \theta (1 - \cos \theta)}.$$ 5. **Combine the fractions:** $$\frac{\sin^2 \theta - \cos \theta (1 - \cos \theta)}{\sin \theta (1 - \cos \theta)}.$$ 6. **Simplify the numerator:** $$\sin^2 \theta - \cos \theta + \cos^2 \theta = (\sin^2 \theta + \cos^2 \theta) - \cos \theta = 1 - \cos \theta.$$ 7. **Substitute back:** $$\frac{1 - \cos \theta}{\sin \theta (1 - \cos \theta)} = \frac{1}{\sin \theta} = \csc \theta.$$ 8. **Conclusion:** The left side simplifies exactly to the right side, so the identity is verified. **Final answer:** $$\frac{\sin \theta}{1 - \cos \theta} - \cot \theta = \csc \theta.$$