Endpoint convergence of a bounded angle diffusion ($d\Theta =\cos \Theta dZ,{\Theta }_{\infty }\in \{-\pi /2,\pi /2\}$)

A diffusion remaining in $(-\pi /2,\pi /2)$ with the displayed equation is a bounded martingale and has a terminal limit. Its expected total quadratic variation is finite by its bounded second moments. An interior limit would leave its squared diffusion coefficient bounded away from zero and force infinite quadratic variation. Thus it converges to an endpoint, and the preserved mean determines the two endpoint probabilities.