Use double angle to find the expression of sin^5(2x)cos(2x)

To find the expression for \(\sin^5(2x)\cos(2x)\) using double angle identities, we can follow these steps:

### Step 1: Express \(\sin^5(2x)\) in terms of \(\sin(2x)\)
\[
\sin^5(2x) = (\sin(2x))^5
\]

### Step 2: Use the identity for \(\sin(2x)\)
\[
\sin(2x) = 2\sin(x)\cos(x)
\]
So, \(\sin^5(2x)\) can be written as:
\[
\sin^5(2x) = (2\sin(x)\cos(x))^5
\]

However, we are interested in using double-angle identities to simplify the product \(\sin^5(2x)\cos(2x)\).

### Step 3: Substitute \(\sin(4x)\) for \(\sin(2x)\cos(2x)\)
The double angle identity for sine and cosine is:
\[
\sin(4x) = 2\sin(2x)\cos(2x)
\]
Therefore, 
\[
\sin(2x)\cos(2x) = \frac{1}{2}\sin(4x)
\]

### Step 4: Substitute into the original expression
Now substitute into the expression \(\sin^5(2x)\cos(2x)\):
\[
\sin^5(2x)\cos(2x) = \sin^4(2x) \cdot \sin(2x) \cdot \cos(2x)
\]

Using the identity \(\sin(2x)\cos(2x) = \frac{1}{2}\sin(4x)\), we get:
\[
\sin^5(2x)\cos(2x) = \sin^4(2x) \cdot \frac{1}{2}\sin(4x)
\]

So the expression becomes:
\[
\sin^5(2x)\cos(2x) = \frac{1}{2}\sin^4(2x)\sin(4x)
\]

### Final Answer:
\[
\sin^5(2x)\cos(2x) = \frac{1}{2}\sin^4(2x)\sin(4x)
\]

Would you like to go into further detail, or do you have any questions?

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Here are some related questions you might find interesting:

1. How would you simplify \(\cos^5(2x)\sin(2x)\) using double angle identities?
2. What is the expression for \(\cos^4(2x)\sin(2x)\cos(2x)\)?
3. How would you find the expression for \(\sin^6(2x)\) using trigonometric identities?
4. What is the double angle identity for \(\sin(2x)\) and how is it derived?
5. How would you express \(\sin^3(2x)\cos(4x)\) using trigonometric identities?
6. What are the identities for \(\sin^2(x)\) and \(\cos^2(x)\) in terms of double angles?
7. How would you express \(\sin^3(x)\cos(x)\) using sum-to-product identities?
8. What is the relationship between \(\sin(4x)\) and \(\sin(2x)\cos(2x)\)?

**Tip:** When dealing with trigonometric identities, always try to simplify the expressions step by step by using fundamental identities like double-angle or sum-to-product formulas.

Learn how to use double angle identities to find the expression for \(\sin^5(2x)\cos(2x)\). This detailed solution covers the application of trigonometric identities, specifically focusing on the double angle identity for sine and cosine. Suitable for high school students and anyone interested in advanced trigonometry problems.

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