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) $x<y<z$
Solve by taking the extreme first. Let $a \rightarrow 0$, then $y \rightarrow 1, z \rightarrow 0$. That is, $y$ is greater than $z$, so we can rule out (B) and (D).
$$
\text { Also, } \frac{x}{z}=(\sin \alpha)^{\log _{6} \sin \alpha-\log _{b} \cos \alpha}=(\sin \alpha)^{\log _{b} 48} \text {. }
$$

Now, take a special case: let $b=\tan a$ (since $0<\tan a<1$, it meets the condition for $b$), then
$$
\frac{x}{y}=\sin \alpha<1 \text {. }
$$

Thus $x<y$. Therefore, the answer is (A).