If A+B+C = pai, prove that
sinA + sinB + sinC = 4 cosA/2 cosB/2 cosC/2
A+B+C = pai
Dividing by 2 on both sides
C/2 = pi/2 - (A/2+B/2)
sinA +sinB +sinC
= 2Sin(A+B)/2 Cos(A-B)/2 + 2SinC/2 Cos C/2
= 2 Sin(pi/2 -C/2)Cos(A-B)/2 + 2SinC/2 Cos C/2
=2 Cos C/2 Cos(A-B)/2 + 2SinC/2 Cos C/2
=2Cos C/2[Cos(A-B)/2 + SinC/2 ]
=2Cos C/2[Cos(A-B)/2 +Sin[pi/2 - (A+B)/2]]
=2Cos C/2[Cos(A-B)/2 +Cos (A+B)/2]
=2Cos C/2[Cos(A/2-B/2) +Cos (A/2+B/2)]
=2Cos C/2[2CosA/2 CosB/2]
=4Cos A/2 Cos B/2 Cos C/2