I was reading a book where I saw the following $$\begin{align}\tilde f(t)&=\frac 1 {\pi}+\frac {\sin t} {2}+\boxed{\frac 1 \pi \sum_{n=2}^\infty \frac {(-1)^{n+1}}{n^2-1}\cos {nt}}\\&=\frac 1 {\pi}+\frac {\sin t} {2}-\boxed{\frac 2 \pi \sum_{n=1}^\infty \frac {1}{4n^2-1}\cos {2nt}}\end{align}$$. I don't understand how the authors get the second box from the first one.
The authors have written "The two series representations for $\tilde f(t)$ are equal because $(-1)^{2k+1}-1=-2$ and $(-1)^{2k}-1=0$."
Any help will be appreciated. Thank you.
Book name: Mathematical methods(2nd ed.), Merle C. Potter, Jack Goldberg.
Page no.379
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