Answer
$-\frac{\sqrt 2}{10}$
Work Step by Step
1. Let $tan^{-1}(-1)=u$ (u in quadrant IV), we have $u=-\frac{\pi}{4}, sin(u)=-\frac{\sqrt 2}{2}$ and $cos(u)=\frac{\sqrt 2}{2}$
2. Let $cos^{-1}(-\frac{4}{5})=v$ (v in quadrant II), we have $cos(v)=-\frac{4}{5}$ and $sin(v)=\frac{3}{5}$
3. Thus $cos(u+v)=cos(u)cos(v)-sin(u)sin(v)=(\frac{\sqrt 2}{2})(-\frac{4}{5})-(-\frac{\sqrt 2}{2})(\frac{3}{5})=-\frac{\sqrt 2}{10}$
