Explanation

Let a = 3 sin(theta) + 5 cos(theta) = 4 and b = 3 cos(theta) - 5 sin(theta). Then a^2 + b^2 = (9 sin^2 + 25 cos^2 + 9 cos^2 + 25 sin^2) = 34, using (a+b)^2 + (a-b)^2 = 2(a^2 + b^2) termwise. So b^2 = 34 - 16 = 18. Hence option D.