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1)cos^6x+sin^6x-sin^2xcos^2x
2)sin^2atga+cos^2actga+sin2a
1)cos^6x+sin^6x-sin^2xcos^2x
2)sin^2atga+cos^2actga+sin2a
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Answers & Comments
cos⁶x+sin⁶x-sin²xcos²x=(cos²x)³+(sin²x)³-sin²xcos²x=
=(cos²x+sin²x)(cos⁴x-sin²xcos²x+sin⁴x)-sin²xcos²x=cos⁴x-2sin²xcos²x+sin⁴x=
=(cos²x-sin²x)²=(cos2x)²=cos²(2x).
2)
sin²atga+cos²actga+sin2a=(1-cos²a)tga+(1-sin²a)ctga+sin2a=
=tga-sinacosa+ctga-sinacosa+sin2a=tga+ctga-2sinacosa+2sinacosa=
=tga+ctga=(sina/cosa)+(cosa/sina)=(sin²a+cos²a)/(sinacosa)=1/(sin2a/2)=
=2/sin2a.