1. **State the problem:** Verify the identity $\tan^2 x \sin^2 x = \tan^2 x - \sin^2 x$ by working only on one side. 2. **Choose the right side to simplify:** We start with the right side: $$\tan^2 x - \sin^2 x$$ 3. **Recall the definition:** $$\tan^2 x = \frac{\sin^2 x}{\cos^2 x}$$ 4. **Rewrite the right side using this:** $$\frac{\sin^2 x}{\cos^2 x} - \sin^2 x$$ 5. **Express $\sin^2 x$ with common denominator:** $$\frac{\sin^2 x}{\cos^2 x} - \frac{\sin^2 x \cos^2 x}{\cos^2 x}$$ 6. **Combine the fractions:** $$\frac{\sin^2 x - \sin^2 x \cos^2 x}{\cos^2 x} = \frac{\sin^2 x (1 - \cos^2 x)}{\cos^2 x}$$ 7. **Use the Pythagorean identity:** $$1 - \cos^2 x = \sin^2 x$$ 8. **Substitute:** $$\frac{\sin^2 x \sin^2 x}{\cos^2 x} = \frac{\sin^4 x}{\cos^2 x}$$ 9. **Recognize this equals the left side:** $$\tan^2 x \sin^2 x = \frac{\sin^4 x}{\cos^2 x}$$ **Final answer:** Simplifying the right side yields the left side, so the identity holds true for all $x$ where $\cos x \neq 0$.