Question

How do I prove that sin(x+y)-sin(x-y)=2cosx siny

Answer 1

just apply the addition formulas:

sin(x+y) = sinx cosy + cosx siny
sin(x-y) = sinx cosy - cosx siny

now subtract. done.

or, apply the sum-to-product formula directly: sinA-sinB = 2sin((A-B)/2)cos((A+B)/2)
so, letting A=x+y and B=x-y

sin(x+y)-sin(x-y) = 2sin(((x+y)-(x-y))/2) cos(((x+y)+(x-y))/2)
= 2 siny cosx

but then, the sum-to-product formula is just based on the addition formulas anyway, so what'd you expect?