I'm guessing this is a trigonometric expansion, just not sure which one.
$\displaystyle 1-x/2+x^2/24-x^3/720+x^4/40320+....$
Notice how the coefficients match those of $\displaystyle \cos x$?
$\displaystyle \cos ({\color{red}x}) = 1 - \frac{{\color{red}x}^2}{2} + \frac{{\color{red}x}^4}{24}- \frac{{\color{red}x}^6}{720} + \frac{{\color{red}x}^8}{40320} + \cdots$
Now think a little and try to make the necessary adjustments to the powers ...
Is the expansion of cos(x) not
$\displaystyle cos(x)=1-x^2/2+x^4/24-x^6/720+x^8/40320+...$ ? (sorry didn't realise you'd edited)
but yeah I was thinking it had something to do with cos, just can't figure it out.
Also I have another expansion I'm interested in...
$\displaystyle 1-x/6+x^2/120-x^3/5040+...$
this has something to do with sin hasn't it?