the question is

Find the Taylor series expansion about x = 0 for cos^x using the series for

2 sin x cos x.

i found the series for 2 sinx cosx by using the sinx since 2 sinx cosx = sin2x

which is $\displaystyle 2x-\frac{4x^3}{3}+\frac{4x^5}{15}+...$

but now im stuck i dont know how 2 sinx cosx can be related to cos^2x