The Prisoner's Dilemma

Since no respondents provided a solution to his last problem, Prof. Aliyev is willing to offer one more chance to unravel his mysterious diagrams. In the accompanying figure, nonintersecting circles with radii have common external tangent lines $AB$ and $CD$, where the named points are the points of tangency. Choose point $M$ on segment $AB$ so that angles $AMD$ and $BMC$ are equal. Let $\theta$ be the angle $DCM$. Show that $\sin \theta \le$ $r/R$.