Note that $(1)$ is true since the LHS equals $\zeta(2)$ whereas the RHS equals $6\arcsin^2 1$ which both turn out to equal $\pi^2/6$ as is well-known. However I am little bit curious about how to show $(1)$ without actually evaluating both series.
I am aware of an elegant approach contributed by Markus Scheuer as an answer to Different methods to compute Basel problem. Although this answers my question partially I am looking for different attemps. For example within Jack D'Aurizio's notes there is a way proposed exploiting creative telescoping $($see page $5$f.$)$ which I am sadly speaking not able to understand completely yet. Hence I have come across a proof of a similiar equality concerning $\zeta(3)$ on AoPS given by pprime I am confident that there are in fact other possible methods.
I would like to see attempts of proving $(1)$ beside the mentioned which do not rely on actually showing that they both equal $\pi^2/6$. Preferably these should be in the spirit of Markus Scheuer's or Jack D'Aurizio's approaches rather than the one similiar by pprime.
I have found another interesting approach, again by Jack D'Aurizio, which can be found here utilizing harmonic sums and creative telescoping in combination.
As pointed out by Zacky on page $31$ of Jack's notes another methode can be found which makes it three possibilities provided by Jack alone. Quite impressive!