3. Given $\alpha, \beta \in\left(\frac{3 \pi}{4}, \pi\right)$,
$$
\cos (\alpha+\beta)=\frac{4}{5}, \sin \left(\alpha-\frac{\pi}{4}\right)=\frac{12}{13} \text {. }
$$

Then $\cos \left(\beta+\frac{\pi}{4}\right)=$ $\qquad$
3. $-\frac{56}{65}$.

From the problem, we have
$$
\sin (\alpha+\beta)=-\frac{3}{5}, \cos \left(\alpha-\frac{\pi}{4}\right)=-\frac{5}{13} .
$$

Then $\cos \left(\beta+\frac{\pi}{4}\right)$
$$
\begin{array}{l}
=\cos (\alpha+\beta) \cdot \cos \left(\alpha-\frac{\pi}{4}\right)+\sin (\alpha+\beta) \cdot \sin \left(\alpha-\frac{\pi}{4}\right) \\
=-\frac{56}{65} .
\end{array}
$$