#how is the average score for this 0.5 out of 10
orchid valley
First, let's simplify the sums.
sin(x)^4 + cos(x)^4 = (sin(x)^4 + cos(x)^4 + 2sin(x)^2 cos(x)^2)) - 2sin(x)^2 cos(x)^2) = (sin(x)^2 + cos(x)^2)^2 - (1/2)(2sin(x)cos(x))^2 = 1 - (1/2)sin(2x)^2
sin(x)^6 + cos(x)^6 = (sin(x)^2 + cos(x)^2)(sin(x)^4 - sin(x)^2 cos(x)^2) + cos(x)^4) = 1 - (1/2)sin(2x)^2 - (1/4)sin(2x)^2 = 1 - (3/4)sin(2x)^2
Thus:
sin(x)^6 + cos(x)^6 + k(sin(x)^4 + cos(x)^4) = 1 - (3/4)sin(2x)^2 + k(1 - (1/2)sin(2x)^2) = k + 1 - (1/4)(2k + 3)sin(2x)^2
The rest should be easy.