sin3Acos(B−C)+sin3Bcos(C−A)+sin3Ccos(A−B)
=sin2AsinAcos(B−C)+sin2BsinBcos(C−A)+sin2CsinCcos(A−B)
=sin2Asin(π−(B+C))cos(B−C)+sin2Bsin(π−(C+A))cos(C−A)+sin2Csin(π−(A+B))cos(A−B) (Using A+B+C=π)
=sin2Asin(B+C)cos(B−C)+sin2Bsin(C+A)cos(C−A)+sin2Csin(A+B)cos(A−B)
(Using sin(π−θ)=sinθ)
=sin2A(sin2B+sin2C)+sin2B(sin2A+sin2C)+sin2C(sin2A+sin2B)2
(Using sinθcosϕ=sin(θ+ϕ)+sin(θ−ϕ)2)
=sin2A2sinBcosB+sin2A2sinCcosC+sin2B2sinAcosA+sin2B2sinCcosC+sin2C2sinAcosA+sin2C2sinBcosB2
(Using sin2θ=2sinθcosθ)
=sinAsinB(sinAcosB+sinBcosA)+sinAsinC(sinAcosC+sinCcosA)+sinBsinC(sinBcosC+sinCcosB)
=sinAsinBsin(A+B)+sinAsinCsin(A+C)+sinBsinCsin(B+C)
(Using sin(θ+ϕ)=sinθcosϕ+cosθsinϕ)
=sinAsinBsin(π−C)+sinAsinCsin(π−B)+sinBsinCsin(π−A)
=3sinAsinBsinC