Example 3. Given $0<b<1,0<a<\frac{\pi}{4}$. Then the size relationship of the following three numbers $x=(\sin \alpha)^{\log _{6} \sin }, y=(\cos \alpha)^{\log _{b} \cos }, z=(\sin \alpha)^{\log _{b} \cos \alpha}$ is ( ).
(A) $x<z<y$
(B) $y<x<x$
(C) $z<x<y$
(D) $x<y<z$