求证|sinx-siny|=|2sin[(x-y)/2]cos[(x+y)/2]|

发布时间：2021-02-19 20:53:51

求证|sinx-siny|=|2sin[(x-y)/2]cos[(x+y)/2]|

网友回答

|2sin[(x-y)/2]cos[(x+y)/2]|
=I2[sin(x/2)cos(y/2)-sin(y/2)cos(x/2)][cos(x/2)cos(y/2)-sin(x/2)(sin(y/2)]
=I2[sin(x/2)cos(x/2)cos²(y/2)-sin²(x/2)sin(y/2)cos(y/2)-cos²(x/2)sin(y/2)cos(y/2)+sin(x/2)cos(x/2)sin²(y/2)]I
=Isinxcos²(y/2)-sin²(x/2)siny-cos²(x/2)siny+sinxsin²(y/2)I
=Isinx[sin²(y/2)+cos²(y/2)]-siny[sin²(x/2)+cos²(x/2)]I
=Isinx-sinyI
得证
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