First, let's use the double angle formula for sine to simplify sin^2(22.5°):

sin^2(22.5°) = (1 - cos(45°))/2
= (1 - √2/2)/2
= (2 - √2)/4

Now, let's simplify sin(22.5°)×cos(22.5°):

sin(22.5°)×cos(22.5°) = √(1 - cos^2(22.5°))×cos(22.5°)
= √(1 - (√2/2)^2)×(√2/2)
= √(1 - 2/4)×√2/2
= √(2/4)×√2/2
= √(1/2)×√2/2
= √(1/2)×√2/2
= 1/√2×√2/2
= 1/2

Therefore, sin²(22.5°) + sin(22.5°)×cos(22.5°) = (2 - √2)/4 + 1/2 = (2 - √2 + 2)/4 = (4 - √2)/4 = 1 - √2/4.