Basically this amounts to finding the maclaurin series for cos(u) since then you can just put in a 2x for u (then multiply by your 1/2 scalar and subtract from 1/2). Note this trick works because the radius of convergence for this series turns out to be infinite, more care should in general be used.
let
f(u)=cos(u) f(0)=1
f'(u)=-sin(u) f'(0)=0
f"(u)=-cos(u) f"(0)=-1
f"'(u)=sin(u) f"'(0)=0
and repeat.
so:
so finally you see:
Think you can take her from here?