Answer
$\frac{2\sqrt {13}(\sqrt {5}-3)}{39}$
Work Step by Step
1. Let $sin^{-1}\frac{2}{3}=u$, we have $sin(u)=\frac{2}{3}, cos(u)=\frac{\sqrt {5}}{3}$
2. Let $tan^{-1}\frac{3}{2}=v$, we have $tan(v)=\frac{3}{2}, sin(v)=\frac{3}{\sqrt {13}}, cos(v)=\frac{2}{\sqrt {13}}$
3. $cos(u+v)=cos(u)cos(v)-sin(u)sin(v)=(\frac{\sqrt {5}}{3})(\frac{2}{\sqrt {13}})-(\frac{2}{3})(\frac{3}{\sqrt {13}})=\frac{2\sqrt {5}-6}{3\sqrt {13}}=\frac{2\sqrt {13}(\sqrt {5}-3)}{39}$
