1. **State the problem:** Prove or simplify the identity: $$\sec^4(x) - \csc^4(x) = \sin^2(x) - \cos^2(x)$$ 2. **Recall definitions and identities:** - $\sec(x) = \frac{1}{\cos(x)}$ - $\csc(x) = \frac{1}{\sin(x)}$ - Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$ 3. **Rewrite the left side using definitions:** $$\sec^4(x) - \csc^4(x) = \frac{1}{\cos^4(x)} - \frac{1}{\sin^4(x)}$$ 4. **Find common denominator and combine:** $$= \frac{\sin^4(x) - \cos^4(x)}{\cos^4(x) \sin^4(x)}$$ 5. **Factor numerator as difference of squares:** $$\sin^4(x) - \cos^4(x) = (\sin^2(x))^2 - (\cos^2(x))^2 = (\sin^2(x) - \cos^2(x))(\sin^2(x) + \cos^2(x))$$ 6. **Use Pythagorean identity in numerator:** $$= (\sin^2(x) - \cos^2(x)) \cdot 1 = \sin^2(x) - \cos^2(x)$$ 7. **Substitute back into fraction:** $$\frac{\sin^4(x) - \cos^4(x)}{\cos^4(x) \sin^4(x)} = \frac{(\sin^2(x) - \cos^2(x))}{\cos^4(x) \sin^4(x)}$$ 8. **Check the right side of the original equation:** The right side is $\sin^2(x) - \cos^2(x)$, which is not equal to the fraction above unless multiplied by $\cos^4(x) \sin^4(x)$. 9. **Conclusion:** The original equation as stated is not an identity unless multiplied by $\cos^4(x) \sin^4(x)$ on the right side. The simplified form of the left side is: $$\sec^4(x) - \csc^4(x) = \frac{(\sin^2(x) - \cos^2(x))}{\cos^4(x) \sin^4(x)}$$ which is different from $\sin^2(x) - \cos^2(x)$ alone. **Final answer:** $$\sec^4(x) - \csc^4(x) = \frac{\sin^2(x) - \cos^2(x)}{\cos^4(x) \sin^4(x)}$$