为什么sin(x+y) - sinx = 2cos(x+y/2)siny/2

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为什么sin(x+y) - sinx = 2cos(x+y/2)siny/2

为什么sin(x+y) - sinx = 2cos(x+y/2)siny/2
为什么sin(x+y) - sinx = 2cos(x+y/2)siny/2

为什么sin(x+y) - sinx = 2cos(x+y/2)siny/2
x+y=x+y/2+y/2
x=x+y/2-y/2
sin(x+y)=sin(x+y/2+y/2)=sin(x+y/2)cosy/2+cos(x+y/2)siny/2
sinx=sin(x+y/2)cosy/2-cos(x+y/2)siny/2
所以
sin(x+y) - sinx
=sin(x+y/2)cosy/2+cos(x+y/2)siny/2-[sin(x+y/2)cosy/2-cos(x+y/2)siny/2]
= 2cos(x+y/2)siny/2

根据积化和差公式:
cosα•sinβ=(1/2)[sin(α+β)-sin(α-β)]
解得:
2cos(x+y/2)siny/2=sin(x+y)-sin(x)

sin(x+y)-sinx=2sin[(x+y)-x]/2*cos[(x+y)+x]/2=2sin(y/2)cos[(2x+y)/2]