Question

化简sin^3α×sin3α+cos^3α×cos3α+1/2sin2α×sin4α

Answer 1

sin^3α应该是(sinα)^3的意思吧，下面的cos^2α=(cosα)^2，等等……
sin2α=2sinαcosα
cos2α=cos^2α-sin^2α
sin3α=sin(α+2α)=sinαcos2α+sin2αcosα=sinα(cos^2α-sin^2α)+2cos^2α×sinα =3sinα×cos^2α-sin^3α
sin4α=sin2(2α)=2sin2αcos2α
cos3α=cos(2α+α)=cos2αcosα-sin2αsinα=(cos^2α-sin^2α)cosα-2sin^2αcosα =cos^3α-3sin^2αcosα

故代入原式=sin^3α×(3sinα×cos^2α-sin^3α)+cos^3α×(cos^3α-3sin^2αcosα)+1/2sin2α×2sin2αcos2α
=3sin^4αcos^2α-sin^6α+cos^6α-3cos^4αsin^2α+cos2α
发现3sin^4αcos^2α-sin^6α+cos^6α-3cos^4αsin^2α=(cos^2α-sin^2α)^3为完全平方式
原式=(cos^2α-sin^2α)^3-cos2α
=（cos2α）^3-cos2α
=cos2α[(cos2α)^2-1]
=cos2α×[-(sin2α)^2]
=-cos2α(sin2α)^2

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Answer 2

sin3α=3sinα-4sin^3α
cos3α=3cosα+4cos^3α
sin^3α×sin3α+cos^3α×cos3α+1/2sin2α×sin4α
=sin^3α×(3sinα-4sin^3α)+cos^3α×(3cosα+4cos^3α)+1/2sin2α×sin4α
=3sin^4α-sin^6α+3cos^4α+cos^6α+2sin^2αcos^4α-2sin^4αcos^2α
=(cos^2α-2sin^2α)^3+3sin^4α+3cos^4α=(cos2α)^3+3sin^4α+3cos^4α
=(cos2α)^3+3(+cos^2α-sin^2α)^2(cos2α)^3+6sin^2αcos^2α
=(cos2α)^3+(cos2α)^2