The problem
I have been trying for a while now to show that this monster $$\begin{align} &\int_0^{\pi/4}\tan(x)\sum_{n=1}^{\infty}(-1)^{n-1}\left(\psi\left(\frac{n}{2}\right)-\psi\left(\frac{n+1}{2}\right)+\frac{1}{n}\right)\sin(2nx)\,\mathrm{d}x \\ +&\int_0^{\pi/4}\cot(x)\sum_{n=1}^{\infty}\left(\psi\left(\frac{n+1}{2}\right)-\psi\left(\frac{n}{2}\right)-\frac{1}{n}\right)\sin(2nx)\,\mathrm{d}x, \end{align} $$ where $\psi$ denotes the Digamma function, is equal to the beautiful Catalan's constant $$G :=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2}.$$ I think it's a great problem; going from such a big expression to a nice, simple constant. We can see that these integrals are very similar, the main difference is probably the $\tan(\cdot)$ and the $\cot(\cdot)$. I think that a crutial step would be to find a Fourier series.
Small thank you note
The user @Quanto helps to higher the quality of this site by posting now and then very interesting integrals. I want to start doing the same. Thank you for inspiring me.