Question

acosA + bcosB + ccosC = 2asinBsinC
prove that lhs = rhs

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