It is verifiable numerically or with a computer algebra system (for example with Mathematica using NIntegrate) that the numerical solution to the following integral
$\displaystyle \int_0^{\frac{\pi }{2}} \cos ^{-1}\left(\frac{1}{2 \cos (x)+1}\right) \, dx$
is equal to the numerical value of $\zeta(2)$ or to $\frac{\pi^2}{6}.$
Could this solution be calculated or determined symbolically or analytically?