9.27 Given $0<b<1,0<a<\frac{\pi}{4}$, then the size relationship of the following three numbers: $x=(\sin \alpha) \log _{b}^{\sin \alpha}, y=(\cos \alpha) \log _{b}^{\cos \alpha}, z=(\sin \alpha) \log _{b}^{\cos \alpha}$ is
(A) $x<z<y$.
(B) $y<\approx<x$.
(C) $z<x<y$.
(D) $x<y<z$.
(China High School Mathematics League, 1994)
[Solution] Given $\because 0\log _{b} \cos \alpha>0$.
$\therefore \quad(\sin \alpha)^{\log _{b} \sin \alpha}<(\sin \alpha)^{\log _{b} \cos \alpha}$, i.e., $x<z$.
Also, $(\sin \alpha)^{\log _{b} \cos \alpha}<(\cos \alpha)^{\log _{b} \cos \alpha}$, i.e., $z<y$.
Therefore, we have $\quad x<z<y$.
Thus, the answer is $(A)$.
Note: This problem can also be solved by taking special values. For example, take $b=\frac{1}{2}, \alpha=\frac{\pi}{6}$.