$\frac{{\pi}^{2}}{{4}^{2}2!}-\frac{{\pi}^{4}}{{4}^{4}4!}+\frac{{\pi}^{6}}{{4}^{6}6!}-\frac{{\pi}^{8}}{{4}^{8}8!}+\cdots $

What I am trying to do is to consider that

$\mathrm{cos}(x)=1-\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}-\frac{{x}^{6}}{6!}+...$

then

$-\mathrm{cos}(x)+1=\frac{{x}^{2}}{2!}-\frac{{x}^{4}}{4!}+\frac{{x}^{6}}{6!}-...$

Up to this point the expression resembles the one I am looking for, however I have not been able to find the final result, any help?