Answer
$\frac{63}{65}$
Work Step by Step
Step 1. Let $u=tan^{-1}\frac{3}{4}, v=cos^{-1}\frac{5}{13}$, we have $tan(u)=\frac{3}{4}, cos(v)=\frac{5}{13}$ and
$sin(u)=\frac{3}{5}, cos(u)=\frac{4}{5}, sin(v)=\frac{12}{13}$ (use the Pythagorean Identities)
Step 2. Use Addition Formula, $sin(tan^{-1}\frac{3}{4}+cos^{-1}\frac{5}{13})=sin(u+v)
=sin(u)cos(v)+cos(u)sin(v)=\frac{3}{5}\times\frac{5}{13}+\frac{4}{5}\times\frac{12}{13}
=\frac{63}{65}$
