To prove this identity, we can start by expanding the left side of the equation using the formula (a + b)^2 = a^2 + 2ab + b^2:

(sin(x) + cos(x))^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x)

Now, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to simplify the above expression:

sin^2(x) + 2sin(x)cos(x) + cos^2(x)
= 1 + 2sin(x)cos(x) (using sin^2(x) + cos^2(x) = 1)
= 1 + sin(x)cos(x) (since 2sin(x)cos(x) = sin(x)cos(x))

Therefore, we have shown that (sin(x) + cos(x))^2 = 1 + sin(x)cos(x).