https://file.aiquickdraw.com/mathbot/file/1730344089947mh30kl2q.jpg

The expression in the image is:

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

To expand this, we use the formula for the square of a binomial, \((a + b)^2 = a^2 + 2ab + b^2\).

Applying this to the expression:

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

Now, using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), we get:

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

So the expression becomes:

\[
1 + 2 \sin x \cos x
\]

We can further simplify \(2 \sin x \cos x\) using the double-angle identity, where \(2 \sin x \cos x = \sin(2x)\):

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

Thus, the simplified form is:

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

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to consider:
1. What are other trigonometric identities that can simplify similar expressions?
2. How would the expansion change if the expression were \((\sin x - \cos x)^2\)?
3. What would be the result of \((\sin x + \cos x)^3\)?
4. How does the double-angle identity help simplify trigonometric expressions?
5. Can we express \((\sin x + \cos x)^2\) in terms of a different trigonometric function?

**Tip:** Knowing the Pythagorean and double-angle identities helps simplify a wide range of trigonometric expressions.

This solution shows how to simplify (sin x + cos x)^2 by expanding the binomial and using trigonometric identities such as the Pythagorean and double-angle identities. Learn to simplify expressions with step-by-step guidance, ideal for Grades 10-12 students.

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