To simplify the expression \((\sin(2x) + \cos(2x))^2\), we can use trigonometric identities and algebraic expansion. 

First, let's expand the square of the binomial:

\[
(\sin(2x) + \cos(2x))^2 = \sin^2(2x) + 2\sin(2x)\cos(2x) + \cos^2(2x)
\]

Next, we apply the Pythagorean identity, which states that \(\sin^2(\theta) + \cos^2(\theta) = 1\):

\[
\sin^2(2x) + \cos^2(2x) = 1
\]

Thus, the expression simplifies to:

\[
1 + 2\sin(2x)\cos(2x)
\]

Using the double angle identity for sine, which is \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\), we can further simplify:

\[
2\sin(2x)\cos(2x) = \sin(4x)
\]

Therefore, the simplified form of the expression is:

\[
1 + \sin(4x)
\]

**Summary:**

The expression \((\sin(2x) + \cos(2x))^2\) simplifies to \(1 + \sin(4x)\).

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Do you have any questions or would you like more details?

Here are some related questions to explore further:
1. What are the Pythagorean identities in trigonometry?
2. How do you use trigonometric identities to simplify expressions?
3. What is the double angle formula for sine?
4. How can you derive the double angle formulas from the sum formulas?
5. What are some applications of trigonometric identities in calculus?
6. How can trigonometric identities be used to solve trigonometric equations?
7. What is the relationship between trigonometric functions and complex numbers?
8. How do you prove trigonometric identities?

**Tip:** Practicing different trigonometric identities helps in solving complex trigonometric equations more easily.

Learn how to simplify the expression (sin(2x) + cos(2x))^2 using trigonometric identities and algebraic techniques. This problem covers the application of the Pythagorean identity and the double angle formula for sine. Suitable for high school students in Grades 10-12.

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