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The given identity to prove is:

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

### Step-by-step Proof:

1. **Expand the left-hand side**:
   We begin by expanding the square of the sum on the left-hand side:
   \[
   (\sin x + \cos x)^2 = (\sin x + \cos x)(\sin x + \cos x)
   \]
   Using the distributive property:
   \[
   = \sin^2 x + 2\sin x \cos x + \cos^2 x
   \]

2. **Simplify using trigonometric identities**:
   We know that \(\sin^2 x + \cos^2 x = 1\) (this is a fundamental Pythagorean identity). So, we substitute \(1\) for \(\sin^2 x + \cos^2 x\):
   \[
   = 1 + 2\sin x \cos x
   \]

Thus, the left-hand side becomes:
\[
1 + 2\sin x \cos x
\]
which is exactly the right-hand side of the equation.

### Conclusion:
The identity is proven.

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

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Here are 5 related questions you can explore:

1. How do you derive the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\)?
2. What are other forms of trigonometric identities involving \(\sin x\) and \(\cos x\)?
3. How does expanding binomials help in solving trigonometric equations?
4. Can you prove other identities using this type of expansion technique?
5. How does the double-angle identity for \(\sin(2x)\) relate to the identity we just proved?

**Tip**: Familiarize yourself with common trigonometric identities—they are key to solving many types of problems efficiently!

Prove the trigonometric identity: (sinx + cosx)^2 = 1 + 2sinx cosx.

Learn how to prove the trigonometric identity (sinx + cosx)^2 = 1 + 2sinx cosx. This problem involves expanding binomials and applying the Pythagorean identity, making it suitable for high school students in Grades 10-12.

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