Simplify cos6θ + sin6θ + 3 cos2θ sin2θ


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Answer:

1

Step by Step Explanation:
  1. Expression can be rewritten as following
    = (cos2θ)3 + (sin2θ)3 + 3 cos2θ sin2θ
  2. Since x3 + y3 = ( x + y ) ( x2 + y2 - xy)
    = (cos2θ + sin2θ ) [ (cos2θ)2 + (sin2θ)2 - cos2θ sin2θ ] + 3 cos2θ sin2θ
  3. Since cos2θ + sin2θ =1 and x2 + y2 = (x+y)2 - 2 xy)
    = (cos2θ + sin2θ ) [ (cos2θ + sin2θ)2 - 2 cos2θ sin2θ - cos2θ sin2θ] + 3 cos2θ sin2θ
    = [ 1 - 3 cos2θ sin2θ ] + 3 cos2θ sin2θ
    = 1

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