I don't think that value is constant. Maybe it should be $\sin A \sin B \cos C$? Let $R$ be the circumradius of $ABC$. Then $CO = R$. $A, B, D, E$ are cyclic so $\angle A = \angle CDE$. By angle chasing we find that $CO \perp DE$ so $CO = CD \sin \angle CDE = CD \sin \angle A$. Next $CD = CA \cos \angle C$, so we have $R = CA \sin \angle A \cos \angle C$. But the extended Law of Sines says $2R = CA / \sin \angle B$. Plugging back and simplifying yields $\sin \angle A \sin \angle B \cos \angle C = 1/2$.

Attachments:

Thanks for you.