5. The size relationship of $\log _{\sin 1} \cos 17 \log _{\sin 1} \operatorname{tg} 1, \log _{\cos 1} \sin 1 \log _{\cos 1} \operatorname{tg} 1$ is
(A) $\log _{\sin 1} \cos 1<\log _{\cos 1} \sin 1<\log _{\sin 1} \operatorname{tg} 1<\log _{\cos 1} \operatorname{tg} 1$;
(B) $\log _{\cos 1} \sin 1<\log _{\cos 1} \operatorname{tg} 1<\log _{\sin 1} \cos 1<\log _{\sin 1} \operatorname{tg} 1$;
(C) $\log _{\sin 1} \operatorname{tg} 1<\log _{\cos 1} \operatorname{tg} 1<\log _{\cos 1} \sin 1<\log _{\sin 1} \cos 1$;
(D) $\log _{\cos 1} \operatorname{tg} 1<\log _{\sin 1} \operatorname{tg} 1<\log _{\sin 1} \cos 1<\log _{\cos 1} \sin 1$.
5. (C)

Given $\frac{\pi}{4}<1<\frac{\pi}{2}$, we have $0<\cos 1<\sin 1<1$, and $\log _{\cos 1} \sin 1>0$, thus we can exclude (A) and (B), leaving only (C) and (D). Furthermore, since $\log _{\cos 1} \sin 1<\log _{\cos 1} \cos 1=\log _{\sin 1} \sin 1<\log _{\sin 1} \cos 1$, we conclude that the correct choice is (C).