For any ΔABC, prove that a+b /c = [cos(A-B/2)]/sin c/2

Question

For any ΔABC, prove that

\(\frac{a+b}{c} = \cfrac{cos(\frac{A-B}{2})}{sin {\frac{C}{2}}}\)

Answer 1

\(\frac{a}{sinA} = \frac{b}{sin B} = \frac{c}{sin C} = k\)

LHS = \(\frac{a+b}{c} = \frac{k\,sinA + k , sin B}{k\, sinC}\)