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 $AX=$
[asy]
draw(Circle((0,0),10));
draw((-10,0)--(8,6)--(2,0)--(8,-6)--cycle);
draw((-10,0)--(10,0));
dot((-10,0));
dot((2,0));
dot((10,0));
dot((8,6));
dot((8,-6));
label("A", (-10,0), W);
label("B", (8,6), NE);
label("C", (8,-6), SE);
label("D", (10,0), E);
label("X", (2,0), NW);
[/asy]
$ \textbf{(A)}\ \cos 6^{\circ}\cos 12^{\circ} \sec 18^{\circ} \qquad\textbf{(B)}\ \cos 6^{\circ}\sin 12^{\circ} \csc 18^{\circ} \qquad\textbf{(C)}\ \cos 6^{\circ}\sin 12^{\circ} \sec 18^{\circ} \\ \qquad\textbf{(D)}\ \sin 6^{\circ}\sin 12^{\circ} \csc 18^{\circ} \qquad\textbf{(E)}\ \sin 6^{\circ} \sin 12^{\circ} \sec 18^{\circ} $