real analysis - Determine the sum $sum_{n=1}^infty left( frac{1}{(4n)^2-1}-frac{1}{(4n+2)^2+1}right)$ using the Fourier series of $|cos x|$

I need to determine the sum $$\sum_{n=1}^\infty \left( \frac{1}{(4n)^2-1}-\frac{1}{(4n+2)^2+1}\right)$$ using the Fourier series of $\lvert \cos x\rvert$ on the interval $[-\pi,\pi]$.

I have already calculated Fourier series and I get this:
$$\lvert \cos x\rvert= {\frac2\pi}+\sum_{n=2}^\infty{\frac4\pi}\frac{\cos(n\frac\pi2)}{1-n^2}\cos(nx)$$

I do not know how to manipulate the Fourier series to get that specific sum.
I tried this but did not get me anywhere.
$$\sum_{n=2}^\infty{\frac4\pi}\frac{{\frac12}\cos(n\frac\pi2-nx)-{\frac12}\cos(n\frac\pi2+nx)}{1-n^2}$$
$$-\sum_{n=2}^\infty{\frac4\pi}\frac{\cos(n\frac\pi2-nx)}{2n^2-2}-\frac{\cos(n\frac\pi2+nx)}{2n^2-2}$$