1. **State the problem:** Prove the identity $$(1 + \tan^2 u)(1 - \sin^2 u) = 1.$$\n\n2. **Recall important formulas:**\n- Pythagorean identity: $$1 + \tan^2 u = \sec^2 u.$$\n- Pythagorean identity: $$1 - \sin^2 u = \cos^2 u.$$\n\n3. **Substitute these identities into the left side:**\n$$ (1 + \tan^2 u)(1 - \sin^2 u) = \sec^2 u \cdot \cos^2 u.$$\n\n4. **Simplify the expression:**\n$$ \sec^2 u \cdot \cos^2 u = \frac{1}{\cos^2 u} \cdot \cos^2 u.$$\n\n5. **Cancel common factors:**\n$$ \frac{\cancel{\cos^2 u}}{\cancel{\cos^2 u}} = 1.$$\n\n6. **Conclusion:** The left side simplifies to 1, which equals the right side, so the identity is proven.