证明sinx+siny+sinz-sin(x+y+z)=4sin((x+y)/2)sin((x+y)/2)sin((x+y)/2)

题目

证明sinx+siny+sinz-sin(x+y+z)=4sin((x+y)/2)sin((x+y)/2)sin((x+y)/2)

答案

sinx+siny+sinz-sin(x+y+z)=4sin[(x+y)/2]sin[(x+z)/2]sin[(y+z)/2]
sinx+siny+sinz-sin(x+y+z)
=2sin[(x+y)/2]cos[(x-y)/2]+sinz-sin(x+y)cosz-sinzcos(x+y)
=2sin[(x+y)/2]cos[(x-y)/2]+sinz[1-cos(x+y)]-sin(x+y)cosz
=2sin[(x+y)/2]cos[(x-y)/2]+2sinz*sin[(x+y)/2]^2-2sin[(x+y)/2]cos[(x+y)/2]cosz
=2sin[(x+y)/2]*{cos[(x-y)/2]+sinzsin[(x+y)/2]-cos[(x+y)/2]cosz}
=2sin[(x+y)/2]*{cos[(x-y)/2]-cos[z+(x+y)/2]}
=2sin[(x+y)/2]*2sin[(x+z)/2]sin[(y+z)/2]
=4sin[(x+y)/2]sin[(x+z)/2]sin[(y+z)/2]