Question

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

Answer 1

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]