求证cosx-cosy=-2sin (x+y/2)*sin (x-y/2)
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求证cosx-cosy=-2sin (x+y/2)*sin (x-y/2)
x=(x+y)/2+(x-y)/2
y=(x+y)/2-(x-y)/2
所以左边=cos[(x+y)/2+(x-y)/2]-cos[(x+y)/2-(x-y)/2]
={cos[(x+y)/2]cos[(x-y)/2]-sin[(x+y)/2]sin[(x-y)/2]}-{cos[(x+y)/2]cos[(x-y)/2]+sin[(x+y)/2]sin[(x-y)/2]}
=-2sin[(x+y)/2]sin[(x-y)/2]
=右边
命题得证
y=(x+y)/2-(x-y)/2
所以左边=cos[(x+y)/2+(x-y)/2]-cos[(x+y)/2-(x-y)/2]
={cos[(x+y)/2]cos[(x-y)/2]-sin[(x+y)/2]sin[(x-y)/2]}-{cos[(x+y)/2]cos[(x-y)/2]+sin[(x+y)/2]sin[(x-y)/2]}
=-2sin[(x+y)/2]sin[(x-y)/2]
=右边
命题得证
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