2. Given $a=\frac{200}{157}$, and $b a^{b}=1$. Then the relationship between $A=\log _{b} \cos \frac{b}{a}$ and $B=\log _{b} \sin \frac{b}{a}$ is ().
(A) $A<B$
(B) $A>B$
(C) $A=B$
(D) Cannot be determined
2.(A).

Given $a>1$, $a^{b}>0$, and $b a^{b}=1>0$, we get $b>0$.
Thus, $a^{b}>1$, and $0<\sin \frac{b}{a}$.
Therefore, $\log _{b} \cos \frac{b}{a}<\log _{b} \sin \frac{b}{a}$.