3. The function that is both odd and monotonically decreasing on the interval $[-1,1]$ is ( ).
(A) $f(x)=\sin x$
(B) $f(x)=-|x+1|$
(C) $f(x)=\ln \frac{2-x}{2+x}$
(D) $f(x)=\frac{1}{2}\left(a^{x}+a^{-x}\right)$
3. C.
If $f(x)=\ln \frac{2-x}{2+x}$, then
$$
f(-x)=\ln \frac{2+x}{2-x}=-\ln \frac{2-x}{2+x}=-f(x) \text {. }
$$

Therefore, $f(x)=\ln \frac{2-x}{2+x}$ is an odd function.
Also, $t=\frac{2-x}{2+x}=-1+\frac{4}{2+x}$ is a monotonically decreasing function on the interval $[-1,1]$, and $y=\ln t$ is a monotonically increasing function on the interval $(0,+\infty)$. Since $f(x)=\ln \frac{2-x}{2+x}$ is the composite function of $y=\ln t$ and $t=\frac{2-x}{2+x}$, the function $f(x)=\ln \frac{2-x}{2+x}$ is monotonically decreasing on the interval $[-1,1]$.