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The image you uploaded presents a series of steps related to solving a mathematical expression involving trigonometric functions. Here’s an explanation of the work shown:

1. **Initial Expression:**
   The given expression is being simplified or integrated:
   \[
   4\theta - \frac{1}{2} \left( 2\sin(2\theta) \cos(2\theta) \right) + \sin(2\theta) - \frac{16}{3} \cos^3(\theta) + C
   \]

2. **Simplification of Trigonometric Terms:**
   - The term \( \frac{1}{2} \left( 2\sin(2\theta) \cos(2\theta) \right) \) simplifies to \( \sin(2\theta) \), because \( 2 \cdot \frac{1}{2} = 1 \). The expression becomes:
     \[
     4\theta + \sin(2\theta) - \frac{16}{3} \cos^3(\theta) + C
     \]
   
3. **Combining Like Terms:**
   - The next part of the equation combines terms involving \( \sin(2\theta) \) into a more compact form:
     \[
     4\theta + \sin(2\theta)(1 - \cos(2\theta)) - \frac{16}{3} \cos^3(\theta) + C
     \]
   - Here, the term \( \sin(2\theta)(1 - \cos(2\theta)) \) is simplified by factoring out \( \sin(2\theta) \).

### Key Steps:

- Trigonometric identities (such as double angle formulas) are used for simplification.
- The expression becomes progressively more manageable, and the terms are grouped appropriately.

Let me know if you'd like further clarification or if you have any questions on specific steps! 

Here are 5 related questions for your learning:
1. How do the double angle formulas for sine and cosine help in simplifying trigonometric expressions?
2. Can you explain how to integrate trigonometric expressions step by step?
3. What other trigonometric identities could be applied to simplify expressions like these?
4. How do constants of integration (like \( C \)) come into play in indefinite integrals?
5. What methods can be used to handle trigonometric integrals that involve higher powers of sine or cosine?

**Tip:** When simplifying trigonometric expressions, always look for opportunities to apply well-known identities like double-angle or power-reduction formulas to make the process smoother.

Explain the steps involved in simplifying and solving the given trigonometric expression.

Learn how to simplify and integrate trigonometric expressions using various trigonometric identities and integration techniques. This problem involves the use of the double angle formula, sine and cosine identities, and the integration of trigonometric functions. Suitable for advanced high school students in Grades 11-12.

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