证明成立：[cos(3x)-sin(3x)]/(cosx+sinx)=1-2sin(2x).

来源：学生作业帮助网编辑：作业帮时间：2024/04/29 07:55:10

证明成立：[cos(3x)-sin(3x)]/(cosx+sinx)=1-2sin(2x).
证明成立：[cos(3x)-sin(3x)]/(cosx+sinx)=1-2sin(2x).

证明成立：[cos(3x)-sin(3x)]/(cosx+sinx)=1-2sin(2x).
cos(3x)-sin(3x)
= cos(2x+x)-sin(2x+x)
= cos2xcosx-sin2xsinx-sin2xcosx-cos2xsinx
= (cos^2x-sin^2x)cosx-2sinxcosxsinx-2sinxcosxcosx-(cos^2x-sin^2x)sinx
= cos^3x-sin^2xcosx-2sin^2xcosx-2sinxcos^2x-cos^2xsinx+sin^3x
= cos^3x+sin^3x-3cos^2xsinx-3cos^2x
= (cosx+sinx)(cos^2x-cosxsinx+sin^2x)-3cosxsinx(cosx+sinx)
= (cosx+sinx)(cos^2x-cosxsinx+sin^2x-3cosxsinx)
= (cosx+sinx)(cos^2x+sin^2x-4cosxsinx)
= (cosx+sinx)(1-2sin2x)
即：cos(3x)-sin(3x) = (cosx+sinx)(1-2sin2x)
∴[cos(3x)-sin(3x)] / (cosx+sinx) = (1-2sin2x)

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