From the given info (and the linked question) we find
[tex]\cos\alpha=-\dfrac5{13}[/tex]
[tex]\sin\alpha=\dfrac{12}{13}[/tex]
[tex]\sin\beta=-\dfrac45[/tex]
Then using the angle-sum identity for cosine, we have
[tex]\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta[/tex]
[tex]\cos(\alpha-\beta)=\left(-\dfrac5{13}\right)\dfrac35+\dfrac{12}{13}\left(-\dfrac45\right)=-\dfrac{63}{65}[/tex]
