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.

Learn how to simplify the trigonometric expression ((1 - cos(2x))/2)^3 ((1 + cos(2x))/2)^2 using identities such as the Pythagorean and double-angle formulas. This step-by-step solution covers key concepts in trigonometry and algebra, suitable for high school students in Grades 10-12.

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