Sure, let's go through the steps to expand the expression [tex]\((\sin \theta + \cos \theta)^2\)[/tex] in detail.
### Step-by-Step Solution
1. Start with the given expression:
[tex]\[
(\sin \theta + \cos \theta)^2
\][/tex]
2. Apply the binomial expansion formula:
[tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]
Here, [tex]\(a = \sin \theta\)[/tex] and [tex]\(b = \cos \theta\)[/tex].
3. Substitute [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] into the binomial expansion formula:
[tex]\[
(\sin \theta + \cos \theta)^2 = (\sin \theta)^2 + 2(\sin \theta)(\cos \theta) + (\cos \theta)^2
\][/tex]
4. Rewrite the expanded terms:
[tex]\[
(\sin \theta)^2 = \sin^2 \theta
\][/tex]
[tex]\[
2(\sin \theta)(\cos \theta) = 2 \sin \theta \cos \theta
\][/tex]
[tex]\[
(\cos \theta)^2 = \cos^2 \theta
\][/tex]
5. Combine all the expanded terms:
[tex]\[
\sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta
\][/tex]
So, the expanded form of [tex]\((\sin \theta + \cos \theta)^2\)[/tex] is:
[tex]\[
\sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta
\][/tex]
This is the detailed step-by-step solution for expanding the given expression.
