Prove trig identity help!

cosA + c(B-C) = 2sinB sinC

If you meant this:
cos(A) + cos(B-C)

Then solve the Left Hand Side. Note that:
cos(A) = -cos(B - C)

So then:
= -cos(B + C) + cos(B - C)

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

= 2sin(B)sin(C)