if A+B+C=π prove that sin(B+2C)+sin(C+2A)+sin(A+2B)=4sin(B-C)/2 sin(C-A)/2 sin(A-B)/2

Answer 1

check out your sum-to-product formulas.

sin(2A+C)+sin(A+2B)
= 2sin((2A+C)/2)cos((A-2B)/2)

C = π-A-B
2A+C = 2A+π-A-B = A-B+π
sin (2A+C)/2 = sin(π/2 + (A-B)/2) = cos(A-B)/2

and similarly for the others. Much stuff will cancel out.