Question

Sinx+cosx=m show that sin^6x+cos^6x=(4-3(m^2-1)^2)/4

Answer 1

Kcos6x+sin6xcos6⁡x+sin6⁡x

A3+B3=(A+B)(A2−AB+B2)A3+B3=(A+B)(A2−AB+B2)

=(cos2x+sin2x)(cos4x−cos2xsin2x+sin4x)=(cos2⁡x+sin2⁡x)(cos4x−cos2⁡xsin2⁡x+sin4⁡x)

cos2x+sin2x=1cos2⁡x+sin2⁡x=1

=(cos4x+2cos2xsin2x+sin4x−3sin2xcos2x)=(cos4⁡x+2cos2⁡xsin2⁡x+sin4⁡x−3sin2⁡xcos2⁡x)

=(cos2x+sin2x)2−3sin2xcos2x=(cos2⁡x+sin2⁡x)2−3sin2⁡xcos2⁡x

=1−3sin2xcos2x


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