1. **State the problem:** Simplify and verify the identity $$(\cot\theta + \csc\theta)(\tan\theta - \sin\theta) = \sec\theta - \cos\theta.$$\n\n2. **Recall definitions and formulas:**\n- $\cot\theta = \frac{\cos\theta}{\sin\theta}$\n- $\csc\theta = \frac{1}{\sin\theta}$\n- $\tan\theta = \frac{\sin\theta}{\cos\theta}$\n- $\sec\theta = \frac{1}{\cos\theta}$\n\n3. **Rewrite the left side using these definitions:**\n$$\left(\frac{\cos\theta}{\sin\theta} + \frac{1}{\sin\theta}\right)\left(\frac{\sin\theta}{\cos\theta} - \sin\theta\right) = \left(\frac{\cos\theta + 1}{\sin\theta}\right)\left(\frac{\sin\theta - \sin\theta \cos\theta}{\cos\theta}\right).$$\n\n4. **Simplify the second factor inside parentheses:**\n$$\frac{\sin\theta - \sin\theta \cos\theta}{\cos\theta} = \frac{\sin\theta(1 - \cos\theta)}{\cos\theta}.$$\n\n5. **Multiply the two factors:**\n$$\frac{\cos\theta + 1}{\sin\theta} \times \frac{\sin\theta (1 - \cos\theta)}{\cos\theta} = \frac{(\cos\theta + 1)(1 - \cos\theta) \sin\theta}{\sin\theta \cos\theta} = \frac{(\cos\theta + 1)(1 - \cos\theta)}{\cos\theta}.$$\n\n6. **Simplify numerator using difference of squares:**\n$$(\cos\theta + 1)(1 - \cos\theta) = 1 - \cos^2\theta = \sin^2\theta.$$\n\n7. **So the expression becomes:**\n$$\frac{\sin^2\theta}{\cos\theta}.$$\n\n8. **Rewrite the right side:**\n$$\sec\theta - \cos\theta = \frac{1}{\cos\theta} - \cos\theta = \frac{1 - \cos^2\theta}{\cos\theta} = \frac{\sin^2\theta}{\cos\theta}.$$\n\n9. **Conclusion:** Both sides equal $$\frac{\sin^2\theta}{\cos\theta}$$ so the identity is verified.