For any triangle ABC, even if B or C is an obtuse angle, a = b cos C + c cos B. Use the Law of Sines to deduce the "addition formula"

sin (B + C) = sin B cos C + sin C cos B

I got most of it but can't figure out the end.

I replaced b with 2R sin B and c with 2R sin C

so

a = 2R sin B cos C + 2R sin C cos B

a = 2R (sin B cos C + sin C cos B)

a/2R = sin B cos C + sin C cos B

sin A = sin B cos C + sin C cos B

I don't know where to go from here