Let $X_1,X_2,...$ be iid random variables with density $(1-\cos x)/\pi x^2$.
How do we show that $$\lim\limits_{n\to\infty}\mathbb{P}\left(\frac{X_1+...+X_n}{n}\leq x\right)=\frac{1}{2}+\pi^{-1}\arctan x?$$
Maybe using the characteristic functions? Then $$\phi_X(u)=\int e^{iux}(1-\cos x)/\pi x^2dx.$$ But how do we proceed?
We have $(1-\cos(x)) = 2\sin^2(x),$ so your PDF is the square of a sinc function, whose Fourier transform is a triangle function $1-|\mu|.$
Since the characteristic function of a sum $n$ of iid variables is the characteristic function to the n-th power, the characteristic function of $S_n/n$ is $$ \phi_{\bar X_n}(\mu) = \left(1-\frac{|\mu|}{n}\right)^n \to e^{-|\mu|}$$
And we know $e^{-|\mu|}$ is the characteristic function of the Cauchy distribution, which is the CDF you have written down.
(To see this is rigorous, one can use Levy's continuity theorem.)