The expression $(\sin 2x)^2$ is the square of $\sin 2x$. We can simplify this expression as:

$(\sin 2x)^2 = \sin^2 2x$

There are no further simplifications unless additional context is provided, such as an identity or a value for $x$.

If you need to express it in terms of a different angle or function, or evaluate it for a particular $x$, let me know!

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you derive the double angle identity for $\sin 2x$?
2. What is $\sin^2 x + \cos^2 x$ equal to?
3. How can you simplify $\sin^2 2x$ using trigonometric identities?
4. How do you differentiate $(\sin 2x)^2$?
5. What are the values of $\sin^2 2x$ for specific angles like $x = \frac{\pi}{4}$?

Tip: The double angle identity for sine is $\sin 2x = 2 \sin x \cos x$, which can often be useful in simplifying trigonometric expressions.