If sin α=

Given:

$\sin \alpha=\frac{4}{5}$

$\cos \beta=\frac{5}{13}$

Now,

$\cos \alpha=\sqrt{1-\sin ^{2} \alpha}=\sqrt{1-\left(\frac{4}{5}\right)^{2}}=\frac{3}{5}$

And,

$\sin \beta=\sqrt{1-\cos ^{2} \alpha}=\sqrt{1-\left(\frac{5}{13}\right)^{2}}=\frac{12}{13}$

Now,

$\cos (\alpha-\beta)=\cos \alpha \times \cos \beta+\sin \alpha \times \sin \beta$

$\Rightarrow \cos (\alpha-\beta)=\frac{3}{5} \times \frac{5}{13} \times \frac{4}{5} \times \frac{12}{13}=\frac{63}{65}$

Thus,

$\cos \frac{\alpha-\beta}{2}=\sqrt{\frac{1+\cos (\alpha-\beta)}{2}}$

$=\sqrt{\frac{1+\frac{63}{65}}{2}}$

$=\frac{8}{\sqrt{65}}$