if f(x)=sin x, show that f(x+y)-f(x)=cos x sin y-2sin x sin^2(y/2)

Answer 1

sin(x+y) = sinx cosy + cosx siny
so, use that to get
f(x+y)-f(x) = sinx cosy + cosx siny - sinx
= cosx siny + sinx (cosy - 1)
Now use the double-angle formula to finish it off.