Let's simplify the expression \((\sin x + \cos x)^2 - 1\).

1. First, expand \((\sin x + \cos x)^2\):

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

2. Next, substitute the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\):

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

3. Now, subtract 1 from the expanded expression:

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

\[
= 2\sin x \cos x
\]

4. Finally, apply the double-angle identity \(\sin 2x = 2\sin x \cos x\):

\[
2\sin x \cos x = \sin 2x
\]

So, the simplified expression is:

\[
\sin 2x
\]

Would you like to see more details or have any questions?

Here are 5 related questions:

1. What is the double-angle identity for cosine?
2. How do you prove the identity \(\sin^2 x + \cos^2 x = 1\)?
3. Can you simplify \((\sin x - \cos x)^2 - 1\)?
4. What is the value of \(\sin 2x\) when \(x = \frac{\pi}{4}\)?
5. How would the expression change if it was \((\sin x + \cos x)^2 + 1\)?

**Tip:** Knowing trigonometric identities, especially the Pythagorean and double-angle identities, is crucial for simplifying expressions effectively.

Learn how to simplify the expression (sin x + cos x)^2 - 1 with a detailed step-by-step solution. This problem involves trigonometric identities, including the Pythagorean identity and double-angle identity. Suitable for high school students studying trigonometry.

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