The value of sin6A + cos6A + 3cos2A sin2A is
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sin6A + cos6A = (sin^2A )^3+ (cos^2A)^3
= (sin^2A + cos^2A) (sin^4A - sin^2A cos^2A +cos^4A) (using identity (a3+ b3= (a+b) (a2 + ab + b2 )
= (1) (sin^2A)^2 +2 sin^2A cos^2A - 2sin^2 A cos^2A -sin^2A cos^2A + cos^4A)
= (sin^2A + cos^2A)^2- 3 sin^2A cos^2A
= 1- 3 sinA^2 cosA^2
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