Given

$\displaystyle \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6... $

$\displaystyle \cos(x) = 1-\frac{1}{2}x^2 + \frac{1}{24}x^4 - \frac{1}{720}x^6 ... $

How would I find the Maclaurin series for $\displaystyle \frac{1}{1+\cos^2(x)}$ ?

I'm pretty sure there's some flaw in my knowledge because I'm not too sure what to do besides let $\displaystyle x = \cos(x) $ which isn't correct.

Could anyone help me out?