Thread: Proof using Law of Sines

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

Since A + B + C = 180 degrees in a triangle. A = 180 - (B + C). Sin A = Sin [180 - (B + C)] = Sin (B + C). Hope this is correct. MHF expert has to vouch this.