jmc

Let $S$ be the set of points on the rays forming the sides of a $120^{\circ }$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are Points $A,B,C$ and $D$ are on a circle of diameter $1$, and $X$ is on diameter $\overline{AD}.$ If $BX=CX$ and $3\angle BAC=\angle BXC=36^{\circ }$, then $AX=$ $$\begin{gathered} \text{(A) }\cos (6^{\circ })\cos (12^{\circ })\sec (18^{\circ }) \\ \text{(B) }\cos (6^{\circ })\sin (12^{\circ })\csc (18^{\circ }) \\ \text{(C) }\cos (6^{\circ })\sin (12^{\circ })\sec (18^{\circ }) \\ \text{(D) }\sin (6^{\circ })\sin (12^{\circ })\csc (18^{\circ }) \\ \text{(E) }\sin (6^{\circ })\sin (12^{\circ })\sec (18^{\circ }) \end{gathered}$$