Question

prove that aCosA+bCosB+cCosC = 2aSinBCosC

Answer 1

we know     a / sin A  =  b / sin B  = c / sin C.
                   b = a cos C + c Cos A
                   c = a Cos B + b cos A

LHS = a cos A + (a cos C + c cos A) cos B + (a cos B + b Cos A) cos C
        = a cos A + cos A (c cos B + b Cos C) + 2 a Cos C Cos B 
        = a cos A  + cos A * a + 2 a cos C cos B
        = 2a [cos A + Cos C Cos B]
        = 2 a [ cos (π-B-C) + cos C cos B]
        = 2 a [- cos (B+C) + cos C cos B]
        = 2 a Sin B sin C