Let's simplify the given expression step by step:

The expression is:

[ \cos\theta \left[\cos\theta \sin\theta - \sin\theta \cos\theta \right] + \sin\theta \left[\sin\theta - \cos\theta \cos\theta \sin\theta \right] ]

Step 1: Simplify each bracket

1. First bracket: $\cos\theta \sin\theta - \sin\theta \cos\theta$ Notice that the two terms are the same but with opposite signs, so: $\cos\theta \sin\theta - \sin\theta \cos\theta = 0$

2. Second bracket: $\sin\theta - \cos\theta \cos\theta \sin\theta$ This can be simplified by factoring $\sin\theta$: $\sin\theta (1 - \cos^2\theta)$ Using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, we can replace $1 - \cos^2\theta$ with $\sin^2\theta$: $\sin\theta \sin^2\theta = \sin^3\theta$

Step 2: Substitute back into the original expression

The expression now becomes: $\cos\theta \cdot 0 + \sin\theta \cdot \sin^3\theta$ Which simplifies to: $0 + \sin^4\theta$

Final Answer

The simplified expression is:

$\sin^4\theta$

Would you like more details or have any questions about this? Here are five related questions you might find interesting:

1. How do you simplify trigonometric expressions involving multiple identities?
2. What is the Pythagorean identity and how is it used in trigonometry?
3. Can you derive the double angle formulas using trigonometric identities?
4. How do you prove trigonometric identities step by step?
5. What are the applications of trigonometric identities in calculus?

Tip: When simplifying trigonometric expressions, always look for opportunities to apply fundamental identities like the Pythagorean identity.