An Application of Vieta's

Let a, b, and c be the sides of the triangle ABC. If a2, b2, and c2 are the roots of the equation x3 -Px2 + Qx - R = 0 (where P, Q, and R are constants), express

cosA/a + cosB/b + cosC/c

In terms of one or more of the coefficients P, Q, and R.

Applying the Law of Cosines to the sides of the triangle, we get

a2 = b2 + c2 - 2bc(cos(A))

Rearranging produces cos(A) = (b2 + c2 - a2)/2bc.

Similarly, on applying the law of cosines to the other sides of the triangle as well we get 

cos(B) = (a2 + c2 - b2)/2ac

cos(C) = (a2 + b2 - c2)/2ab

From these substitutions, it suffices to find the value of

(b2 + c2 - a2)/2abc + (a2 + c2 - b2)/2abc + (a2 + b2 - c2)/2abc = (a2 + b2 + c2)/2abc (*)

Applying Vieta’s formulas to the given polynomial, P = a2 + b2 + c2 and R = (abc)2 ⇒ √R = abc.

Finally, we complete the problem by applying the given observation to the target value (*) -

P/2√R