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, b = k sin B, c = k sin C

Now,

Substituting the values from sine rule into the above equation, we get

But

And

And

Substituting these we get

By canceling the like terms, we get

Similarly,

Substituting the values from sine rule into the above equation, we get

But

And

And

Substituting these we get

By canceling the like terms, we get

Similarly,

Substituting the values from sine rule into the above equation, we get

But

And

And

Substituting these we get

By canceling the like terms, we get

So the LHS of the given equation, we get

From equation (i), (ii) and (iii), we get

Hence proved