Ross's Introduction to Stochastic Processes states, but does not prove this result:
For Brownian motion, let $A(t)$ denote the amount of time in $[0,t]$ the process is positive. Then, for $0<x<1$, $$\mathbb P(A(t)/t\leqslant x) = \frac2\pi \arcsin \sqrt x. $$
How can this be proven?