1. **State the problem:** Prove or disprove the identity $$\tan^2 \theta - \sin^2 \theta \stackrel{?}{=} \sin^2 \theta \tan^2 \theta$$. 2. **Recall definitions and formulas:** - $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ - Therefore, $$\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}$$. 3. **Rewrite the left side (LHS):** $$\tan^2 \theta - \sin^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta$$ 4. **Find a common denominator:** $$= \frac{\sin^2 \theta}{\cos^2 \theta} - \frac{\sin^2 \theta \cos^2 \theta}{\cos^2 \theta}$$ 5. **Combine the fractions:** $$= \frac{\sin^2 \theta - \sin^2 \theta \cos^2 \theta}{\cos^2 \theta}$$ 6. **Factor out $$\sin^2 \theta$$ in the numerator:** $$= \frac{\sin^2 \theta (1 - \cos^2 \theta)}{\cos^2 \theta}$$ 7. **Use the Pythagorean identity:** $$1 - \cos^2 \theta = \sin^2 \theta$$ 8. **Substitute:** $$= \frac{\sin^2 \theta \cdot \sin^2 \theta}{\cos^2 \theta} = \frac{\sin^4 \theta}{\cos^2 \theta}$$ 9. **Rewrite the right side (RHS):** $$\sin^2 \theta \tan^2 \theta = \sin^2 \theta \cdot \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\sin^4 \theta}{\cos^2 \theta}$$ 10. **Compare LHS and RHS:** Both sides equal $$\frac{\sin^4 \theta}{\cos^2 \theta}$$. **Conclusion:** The identity is true because both sides simplify to the same expression. **Final answer:** $$\tan^2 \theta - \sin^2 \theta = \sin^2 \theta \tan^2 \theta$$