We aim to simplify the given expression:

$\left(\cos^2x\right)^2 - \left(\sin^2x\right)^2 - \cos^2x \sin^2x$

Step 1: Rewrite $(\cos^2x)^2$ and $(\sin^2x)^2$

$(\cos^2x)^2 = \cos^4x \quad \text{and} \quad (\sin^2x)^2 = \sin^4x$

Substitute back: $\cos^4x - \sin^4x - \cos^2x \sin^2x$

Step 2: Recognize the difference of squares in $\cos^4x - \sin^4x$

Using the identity $a^2 - b^2 = (a - b)(a + b)$, we get: $\cos^4x - \sin^4x = (\cos^2x - \sin^2x)(\cos^2x + \sin^2x)$

But $\cos^2x + \sin^2x = 1$ (Pythagorean identity), so: $\cos^4x - \sin^4x = \cos^2x - \sin^2x$

Now the expression becomes: $(\cos^2x - \sin^2x) - \cos^2x \sin^2x$

Step 3: Simplify further

The expression is: $\cos^2x - \sin^2x - \cos^2x \sin^2x$

Factorize if possible: $\cos^2x - \sin^2x - \cos^2x \sin^2x = (\cos^2x - \sin^2x)(1 - \cos^2x \sin^2x)$

Final Simplified Form:

The simplified expression is: $(\cos^2x - \sin^2x)(1 - \cos^2x \sin^2x)$

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Would you like me to verify this result or explore specific parts in more depth?

Here are 5 follow-up questions to consider:

1. What are common trigonometric identities used in simplifications?
2. Can we rewrite $1 - \cos^2x \sin^2x$ using other identities?
3. How is the difference of squares principle applied in trigonometry?
4. Can the final result be expressed in terms of $\sin 2x$ or $\cos 2x$?
5. How do such expressions relate to symmetry in trigonometric graphs?

Tip: Always remember the Pythagorean identity, $\cos^2x + \sin^2x = 1$, as it often simplifies trigonometric problems significantly.