Simplify sin6β + cos6β + 3sin2β cos2β.


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Answer:

1

Step by Step Explanation:
  1. S = sin6β + cos6β + 3sin2β cos2β
  2. ⇒ S = (sin2β)3 + (cos2β)3 + 3sin2β cos2β
  3. ⇒ S = (sin2β + cos2β) [(sin2β)2 - sin2β cos2β + (cos2β)2] + 3sin2β cos2β
    .................. Using a3+b3 = (a+b)(a2 - ab + b2)
  4. ⇒ S = 1[(sin2β)2 - sin2β cos2β + (cos2β)2] + 3sin2β cos2β
    .................. Using sin2θ + cos2θ = 1
  5. ⇒ S = (sin2β)2 + 2sin2β cos2β + (cos2β)2
  6. ⇒ S = (sin2β + cos2β)2
    .................. Using a2 + b2 + 2ab = (a + b)2
  7. ⇒ S = 12
    .................. Using sin2θ + cos2θ = 1
    ⇒ S = 1

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