You want to simplify the expression $\left( \frac{1 - \cos(2x)}{2} \right)^3 \left( \frac{1 + \cos(2x)}{2} \right)^2$

Step-by-step simplification:

1. The expression can be written as: $\frac{(1 - \cos(2x))^3}{2^3} \cdot \frac{(1 + \cos(2x))^2}{2^2}$ Which simplifies to: $\frac{(1 - \cos(2x))^3 (1 + \cos(2x))^2}{2^5}$

2. Now observe the product $(1 - \cos(2x))^3 (1 + \cos(2x))^2$. To simplify it further, let's break it down:

   • We know that $(1 - \cos(2x))(1 + \cos(2x)) = 1 - \cos^2(2x)$.
   • So, the expression becomes: $\frac{(1 - \cos^2(2x)) (1 - \cos(2x))^2}{2^5}$
   • Since $1 - \cos^2(2x) = \sin^2(2x)$, we get: $\frac{\sin^2(2x) (1 - \cos(2x))^2}{32}$

Thus, the simplified expression is: $\frac{\sin^2(2x) (1 - \cos(2x))^2}{32}$

Final Answer:

$\frac{\sin^2(2x) (1 - \cos(2x))^2}{32}$

Let me know if you'd like to explore this further or have any questions!

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Here are 5 questions to expand the topic:

1. How does the identity $\sin^2(2x) = 1 - \cos^2(2x)$ help in simplifying expressions?
2. What are the common trigonometric identities used in simplifying similar expressions?
3. Can this expression be simplified further for specific values of $x$?
4. How does the symmetry of sine and cosine functions influence expression simplifications?
5. How does knowing basic trigonometric identities help in solving calculus problems?

Tip: Trigonometric identities like $\sin^2(x) + \cos^2(x) = 1$ and the double-angle formulas are powerful tools for simplifying complex expressions.