Question

PROOF THAT,
sin(A)+sin(B)-sin(C) = 4sin(A/2) . sin(B/2) . sin(C/2)

Answer 1

Question is mentioned wrongly.  it should be cos C/2 and not sin C/2.

A+B+C = 180
Sin A+ Sin B - Sin C = SIN A + Sin B - Sin (180  - (A+B) )
    = SIN A + SIN B - SIN (A+B)
    = SIN A + SIN B - SIN A COS B + COS A SIN B
    = Sin A ( 1 - Cos B ) + Sin B (1 - Cos A)
    =  2 Sin A/2 Cos A/2 * 2 Sin² B/2 + 2 Sin B/2 Cos B/2 * 2 Sin² A/2
   =  4 Sin A/2 Sin B/2 [ sin B/2 Cos A/2 + Cos B/2  Sin A/2 ]
   = 4 Sin A/2 COs B/2 Sin (B+A)/2 
  = 4 Sin A/2 Cos B/2  Sin (180 - C)/2
   = 4 Sin A/2 Cos B/2  Cos C/2