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[tex]A=x\\\\sin5x=5cos^4xsinx-10cos^2xsin^3x+sin^5x\\\\L=sin(3x+2x)=sin3xcos2x+sin2xcos3x=(*)\\-------------------------------\\sin3x=sin(2x+x)=sin2xcosx+sinxcos2x\\=2sinxcosxcosx+sinx(cos^2x-sin^2x)\\=2sinxcos^2x+sinxcos^2x-sin^3x=3sinxcos^2x-sin^3x\\-------------------------------[/tex]
[tex]sin3xcos2x=(3sinxcos^2x-sin^3x)(cos^2x-sin^2x)\\=3sinxcos^4x-3sin^3xcos^2x-sin^3xcos^2x+sin^5x\\=3sinxcos^4x-4sin^3xcos^2x+sin^5x\\-------------------------------[/tex]
[tex]cos3x=cos(2x+x)=cos2xcosx-sin2xsinx\\=cosx(cos^2x-sin^2x)-2sinxcosxsinx\\=cos^3x-sin^2xcosx-2sin^2xcosx=cos^3x-3sin^2xcosx\\-------------------------------[/tex]
[tex]sin2xcos3x=2sinxcosx(cos^3x-3sin^2xcosx)\\=2sinxcos^4x-6sin^3xcos^2x\\-------------------------------[/tex]
[tex] (*)=3sinxcos^4x-4sin^3xcos^2x+sin^5x+2sinxcos^4x-6sin^3xcos^2x\\\\=5sinxcos^4x-10sin^3xcos^2x+sin^5x=R[/tex]
[tex]=======================================\\\\sin(\alpha+\beta)=sin\alpha cos\beta+sin\beta cos\alpha\\\\cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta\\\\sin2\alpha=2sin\alpha cos\alpha\\\\cos\alpha=cos^2\alpha-sin^2\alpha[/tex]
[tex]sin3xcos2x=(3sinxcos^2x-sin^3x)(cos^2x-sin^2x)\\=3sinxcos^4x-3sin^3xcos^2x-sin^3xcos^2x+sin^5x\\=3sinxcos^4x-4sin^3xcos^2x+sin^5x\\-------------------------------[/tex]
[tex]cos3x=cos(2x+x)=cos2xcosx-sin2xsinx\\=cosx(cos^2x-sin^2x)-2sinxcosxsinx\\=cos^3x-sin^2xcosx-2sin^2xcosx=cos^3x-3sin^2xcosx\\-------------------------------[/tex]
[tex]sin2xcos3x=2sinxcosx(cos^3x-3sin^2xcosx)\\=2sinxcos^4x-6sin^3xcos^2x\\-------------------------------[/tex]
[tex] (*)=3sinxcos^4x-4sin^3xcos^2x+sin^5x+2sinxcos^4x-6sin^3xcos^2x\\\\=5sinxcos^4x-10sin^3xcos^2x+sin^5x=R[/tex]
[tex]=======================================\\\\sin(\alpha+\beta)=sin\alpha cos\beta+sin\beta cos\alpha\\\\cos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta\\\\sin2\alpha=2sin\alpha cos\alpha\\\\cos\alpha=cos^2\alpha-sin^2\alpha[/tex]
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