$\sin \alpha+\sin \beta=-\frac{21}{65}$ and $\cos \alpha+\cos \beta=-\frac{27}{65}$.
Squaring and adding, we get
$$
\begin{aligned}
& 2+2 \cos (\alpha-\beta)=\frac{1170}{(65)^2} \\
& \Rightarrow \cos ^2\left(\frac{\alpha-\beta}{2}\right)=\frac{9}{130} \Rightarrow \cos \left(\frac{\alpha-\beta}{2}\right)=\frac{-3}{\sqrt{130}} \quad\left(\because \frac{\pi}{2} < \frac{\alpha-\beta}{2} < \frac{3 \pi}{2}\right) .
\end{aligned}
$$