1. Let $\alpha \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$, then the order of $(\cos \alpha)^{\cos \alpha},(\sin \alpha)^{\cos \alpha},(\cos \alpha)^{\sin \alpha}$ is .
(A) $(\cos \alpha)^{\cos \alpha}<(\sin \alpha)^{\cos \alpha}<(\cos \alpha)^{\sin \alpha}$;
(B) $(\cos \alpha)^{\cos \alpha}<(\cos \alpha)^{\sin \alpha}<(\sin \alpha)^{\cos \alpha}$;
(C) $(\sin \alpha)^{\cos \alpha}<(\cos \alpha)^{\cos \alpha}<(\cos \alpha)^{\sin \alpha}$;
(D) $(\cos \alpha)^{\sin \alpha}<(\cos \alpha)^{\cos \alpha}<(\sin \alpha)^{\cos \alpha}$.
1. (D)
$$
\begin{array}{ll}
\because & \alpha \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right), \\
\therefore & 0<\cos \alpha<\frac{\sqrt{2}}{2}<\sin \alpha<1 \\
\text { Therefore } & \\
\therefore & (\cos \alpha)^{\sin \alpha}<(\cos \alpha)^{\cos \alpha}, \\
(\cos \alpha)^{\cos \alpha}<(\sin \alpha)^{\cos \alpha} .
\end{array}
$$