Your polynomial is correct; see WolframAlpha.
i cant believe i got this wrong but I did and I dont know why.
find the 2nd order Taylor Polynomial of centered at zero
f(x) = cos x , then f(0) = 1
f'(x) = -sinx, f'(0) =0
f"(x) = -cosx f"(0) = -1
second order TP of cosx is:
Replace x with x^2, second order TP of :
But my book says that P"(x) = 1