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.