#A+B+C=pi rArr A+B=pi-C.#
#:. A+2B=(A+B)+B=(pi-C)+B=pi-(C-B).#
#:. sin(A+2B)=sin(pi-(C-B))=sin(C-B).#
Similarly,
# sin(B+2C)=sin(A-C), and, sin(C+2A)=sin(B-A).#
#:. sin(A+2B)+sin(B+2C)+sin(C+2A),#
#=sin(C-B)+sin(A-C)+sin(B-A).#
Let, #C-B=x, A-C=y and B-A=z.#
Then, #x+y+z=0.#
#:. x+y=-z......(1) and z=-(x+y)......(2).#
#"The L.H.S.="sinx+siny+sinz,#
#=2sin((x+y)/2)cos((x-y)/2)+sinz,#
#=2sin(-z/2)cos((x-y)/2)+sinz,............[because (1)],#
#=-2sin(z/2)cos((x-y)/2)+2sin(z/2)cos(z/2),...........................................................................[because sin2a=2sinacosa],#
#=2sin(z/2){cos(z/2)-cos((x-y)/2)},#
#=2sin(z/2){cos(-(x+y)/2)-cos((x-y)/2)},.......[because (2)],#
#=2sin(z/2){cos((x+y)/2)-cos((x-y)/2)},#
#=2sin(z/2){-2sin(x/2)sin(y/2)},#
#=-4sin(x/2)sin(y/2)sin(z/2),#
#=-4sin((C-B)/2)sin((A-C)/2)sin((B-A)/2),#
#=-4sin(-(B-C)/2)sin(-(C-A)/2)sin(-(A-B)/2),#
#=+4sin((B-C)/2)sin((C-A)/2)sin((A-B)/2),#
#"=The R.H.S."#
Enjoy Maths.!