$$
\begin{array}{l}
\tan 1>1>\sin 1>\cos 1>0 \Rightarrow \log _{\sin 1} \tan 1, \log _{\cos 1} \tan 1\log _{\sin 1} \sin 1=1,0<\log _{\cos 1} \sin 1=\frac{1}{\log _{\sin 1} \cos 1}<1,
\end{array}
$$

Combining the options, we know that the answer is $C$.
5. The size relationship of $\log _{\sin 1} \cos 1, \log _{\sin 1} \tan 1, \log _{\cos 1} \sin 1, \log _{\cos 1} \tan 1$ is
A. $\log _{\sin 1} \cos 1<\log _{\cos 1} \sin 1<\log _{\sin 1} \tan 1<\log _{\cos 1} \tan 1$
B. $\log _{\cos 1} \sin 1<\log _{\cos 1} \tan 1<\log _{\sin 1} \cos 1<\log _{\sin 1} \tan 1$
C. $\log _{\sin 1} \tan 1<\log _{\cos 1} \tan 1<\log _{\cos 1} \sin 1<\log _{\sin 1} \cos 1$
D. $\log _{\cos 1} \tan 1<\log _{\sin 1} \tan 1<\log _{\sin 1} \cos 1<\log _{\cos 1} \sin 1$