In any triangle ABC, prove the following:

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 the above equation, we get

Applying half angle rule,

And

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

But

And

So the above equations in equation (v), we get

Dividing numerator and denominator by, we get

By canceling the like terms we get,

Hence proved