The expression in the image is:

$(\sin \phi + \cos \phi)^2$

Let’s expand and simplify it.

Step 1: Use the identity for a binomial square

$(a + b)^2 = a^2 + 2ab + b^2$ Here, $a = \sin \phi$ and $b = \cos \phi$.

Step 2: Apply the identity to the expression

$(\sin \phi + \cos \phi)^2 = \sin^2 \phi + 2\sin \phi \cos \phi + \cos^2 \phi$

Step 3: Use the Pythagorean identity

We know that: $\sin^2 \phi + \cos^2 \phi = 1$

Thus: $\sin^2 \phi + 2\sin \phi \cos \phi + \cos^2 \phi = 1 + 2\sin \phi \cos \phi$

Final Simplified Expression

$(\sin \phi + \cos \phi)^2 = 1 + 2\sin \phi \cos \phi$

This is the simplified form.

Do you have any questions or need further details?

Related Questions:

1. How can the identity $\sin^2 \phi + \cos^2 \phi = 1$ be derived?
2. What are other useful trigonometric identities?
3. Can the expression $1 + 2\sin \phi \cos \phi$ be further factored?
4. What happens if $\phi$ takes specific values, such as $0^\circ$ or $45^\circ$?
5. How can we rewrite $2\sin \phi \cos \phi$ in terms of a double angle?

Tip: The expression $2\sin \phi \cos \phi$ can also be rewritten using the double angle identity: $2\sin \phi \cos \phi = \sin(2\phi)$.