We are asked to prove that (sin x + cos x)^2 = 1 + 2 sin(x) cos(x).
1) Change (sin x + cos x)^2 to (sin x + cos x)(sin x + cos x) (since the square of any expression is that expression multiplied by itself.)
2) Utilize the FOIL method for multiplying binomials, e.g. (sin x + cos x)(sin x + cos x) = (sin x)(sin x) + (sin x)(cos x) + (cos x)(sin x) + (cos x)(cos x)
3) Simplify and group like terms: (sin x)(sin x) + (sin x)(cos x) + (cos x)(sin x) + (cos x)(cos x) = sin^2 x + cos^2 x + 2 sin x cos x
4) Recall the trigonometric identity which states sin^2 x + cos ^2 x =1, and substitute into (3): sin^2 x + cos ^2 x + 2 sin x cos x = 1 + 2 sin x cos x
5) Use substitution: (sin x + cos x)^2 = 1 + 2 sin x cos x
