Answer
$\frac{63}{65}$
Work Step by Step
Step 1. Letting $sin^{-1}(\frac{5}{13})=u$, we have $sin(u)=\frac{5}{13}$, thus $cos(u)=\frac{12}{13}$
Step 2. Letting $tan^{-1}(\frac{3}{4})=v$, we have $tan(v)=\frac{3}{4}$, thus $sin(v)=\frac{3}{5},cos(v)=\frac{4}{5}$
Step 3. $cos(u-v)=cos(u)cos(v)+sin(u)sin(v)=(\frac{12}{13})(\frac{4}{5})+(\frac{5}{13})(\frac{3}{5})=\frac{48}+15{65}=\frac{63}{65}$
