Answer
$-\frac{33}{65}$
Work Step by Step
Step 1. Letting $tan^{-1}(\frac{4}{3})=u$, we have $tan(u)=\frac{4}{3}$, thus $sin(u)=\frac{4}{5},cos(u)=\frac{3}{5}$
Step 2. Letting $cos^{-1}(\frac{5}{13})=v$, we have $cos(v)=\frac{5}{13}$, thus $sin(v)=\frac{12}{13}$
Step 3. $cos(u+v)=cos(u)cos(v)-sin(u)sin(v)=(\frac{3}{5})(\frac{5}{13})-(\frac{4}{5})(\frac{12}{13})=\frac{15-48}{65}=-\frac{33}{65}$
