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.!