Answer
$\frac{63}{65}$
Work Step by Step
1. Let $sin^{-1}(\frac{5}{13})=u$ (quadrant I), we have $sin(u)=\frac{5}{13}, cos(u)=\frac{12}{13}$
2. Let $tan^{-1}(\frac{3}{4})=v$ (quadrant I), we have $tan(v)=\frac{3}{4}, sin(v)=\frac{3}{5}, cos(v)=\frac{4}{5}, $
3. $cos(u-v)=cos(u)cos(v)+sin(u)sin(v)=(\frac{12}{13})(\frac{4}{5})+(\frac{5}{13})(\frac{3}{5})=\frac{63}{65}$