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.