Q.17 In \(\Delta ABC\), prove that : \(\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{a^2 + b^2 + c^2}{2abc}\).
Best answer
Answer: Substitute the cosine rule for each angle and add.
By the cosine rule, \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\), so \(\frac{\cos A}{a} = \frac{b^2 + c^2 - a^2}{2abc}\).
Similarly \(\frac{\cos B}{b} = \frac{c^2 + a^2 - b^2}{2abc}\) and \(\frac{\cos C}{c} = \frac{a^2 + b^2 - c^2}{2abc}\).
L.H.S. \(= \frac{(b^2 + c^2 - a^2) + (c^2 + a^2 - b^2) + (a^2 + b^2 - c^2)}{2abc} = \frac{a^2 + b^2 + c^2}{2abc}\) = R.H.S.
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