求证!cos^8x-sin^8x+1/4sin2xsin4x =cos2x

题目

求证!cos^8x-sin^8x+1/4sin2xsin4x =cos2x
求数学大神帮忙详细过程 谢谢

答案

cos^8x-sin^8x+1/4sin2xsin4x-cos2x
=(cos^4x+sin^4x)(cos^4x-sin^4x)+1/2sin²2xcos2x-cos2x
=(cos^4x+sin^4x)(cos²x-sin²x)+2sin²xcos²x(cos²x-sin²x)-cos2x
=cos^6x-sin^6x+sin²xcos^4x-sin^4xcos²x-cos2x
=cos^4x-sin^4x-cos2x
=cos²x-sin²x-cos2x
=cos2x-cos2x
=0
所以cos^8x-sin^8x+1/4sin2xsin4x-cos2x=0
cos^8x-sin^8x+1/4sin2xsin4x =cos2x