Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get

⇒a = k sin A

Similarly, b = k sin B

And c = k sin C…..(i)

So, a + b = k(sin A + sin B)..(ii)

So the given LHS becomes,

Substituting equation (i) and (ii) in above equation, we get

Applying half angle rule,

And

Substituting equation (iii) and (iv) in equation (ii), we get

Hence proved