To simplify the expression [tex]\((\sin x - \cos x)^2 + \sin 2x\)[/tex], we will proceed step-by-step:
1. Expand [tex]\((\sin x - \cos x)^2\)[/tex]:
[tex]\[
(\sin x - \cos x)^2 = (\sin x - \cos x)(\sin x - \cos x)
\][/tex]
Applying the distributive property (FOIL method):
[tex]\[
= \sin^2 x - \sin x \cos x - \sin x \cos x + \cos^2 x
\][/tex]
Simplify by combining like terms:
[tex]\[
= \sin^2 x + \cos^2 x - 2 \sin x \cos x
\][/tex]
2. Use the Pythagorean identity:
Recall the Pythagorean identity:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
So, substitute [tex]\(1\)[/tex] in the expression:
[tex]\[
(\sin x - \cos x)^2 = 1 - 2 \sin x \cos x
\][/tex]
3. Write [tex]\(\sin 2x\)[/tex] in terms of [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex]:
Recall the double-angle identity for sine:
[tex]\[
\sin 2x = 2 \sin x \cos x
\][/tex]
4. Substitute [tex]\(\sin 2x\)[/tex] into the expression:
So the original expression becomes:
[tex]\[
1 - 2 \sin x \cos x + \sin 2x
\][/tex]
Substitute [tex]\(\sin 2x\)[/tex] with [tex]\(2 \sin x \cos x\)[/tex]:
[tex]\[
1 - 2 \sin x \cos x + 2 \sin x \cos x
\][/tex]
5. Simplify the expression:
Notice that [tex]\(- 2 \sin x \cos x\)[/tex] and [tex]\(2 \sin x \cos x\)[/tex] cancel each other out:
[tex]\[
1 - 2 \sin x \cos x + 2 \sin x \cos x = 1
\][/tex]
Therefore, the simplified form of the expression [tex]\((\sin x - \cos x)^2 + \sin 2x\)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
