I've consider the following calculations from the so-called Wilker's Inequalities, that I've found in MathWorld, where is mentioned [1] that I add as reference*:
Divide by $\tan(x)$, perform the specialization $x=\frac{1}{n}$ for integers $n\geq 1$, after multiply the inequality by $\frac{1}{n^4}$ and take the sum $\sum_{n=1}^{\infty}$ to state the resulting inequality.
My problem is that I can to prove computationally that $\operatorname{Left side term}<\operatorname{Middle term}$, but my calculations provide me the statement $$\operatorname{Right side term}<\operatorname{Middle term}.$$
*The purpose of previous paragraph is just to provide motivation and context to my question. Isn't required thus nothing about the mentioned inequality that I should to study in the library of my university.
Question. Do you know if were in the literature closed-form or good approximations of the series $$\sum_{n=1}^\infty\frac{\sin\left(\frac{1}{n}\right)\cos\left(\frac{1}{n}\right)}{n^2}\tag{1}$$ and/or $$\sum_{n=1}^\infty\frac{\cos\left(\frac{1}{n}\right)}{n^4\sin\left(\frac{1}{n}\right)}?\tag{2}$$ Then answer this question as a reference request, and I try to find and read those justifications (I don't know if these series are standard to calculate and justify a good approximation). Many thanks.
References:
[1] Borwein, Bailey, and Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters/CRC Press (2004).