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$\sum_{k=1}^\infty\frac{\sin(k)\cos(k)}{k^3}=\frac{\pi}{4}$

How to prove that the sum of the following series is equal to $\frac{\pi}{4}$?
$$\sum_{k=1}^\infty\frac{\sin(k)\cos(k)}{k^3}=\frac{\pi}{4}$$
I tried to differentiate and calculate
$$\sum_{k=1}^\infty\frac{\sin(k)\cos(k)}{k^3}=\frac{d}{dk}\frac{\sin(k)\cos(k)}{k}=\frac{d}{dk}\left(\frac{\sin(k)\cos(k)}{k+\sin(k)\cos(k)}-\frac{2}{k}\right)$$
but I do not know what to do next.

A:

The sum is equal to $\dfrac{d}{dk}\left(\sum_{n=1}^\infty \dfrac{k\cos^2(k)}{n^2}-\dfrac{2}{k}\right)$.

Now, note that
\sum_{n=1}^\infty \dfrac{k\cos^2

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