$\eqalign{
  & \operatorname{Cos} \alpha  = \frac{4}{5} \to {\operatorname{Sin} ^2}\alpha  = 1 - {\operatorname{Cos} ^2}\alpha  = 1 - \frac{{16}}{{25}} = \frac{9}{{25}} \to \operatorname{Sin} \alpha  =  \pm \frac{3}{5} \to \operatorname{Sin} \alpha  = \frac{3}{5}  \cr 
  & \operatorname{Cos} \beta  = \frac{{ - 12}}{{13}} \to {\operatorname{Sin} ^2}\beta  = 1 - {\operatorname{Cos} ^2}\beta  = 1 - \frac{{144}}{{169}} = \frac{{25}}{{169}} \to \operatorname{Sin} \beta  =  \pm \frac{5}{{13}}  \cr 
  & \operatorname{Sin} (\alpha  + \beta ) = \operatorname{Sin} \alpha .\operatorname{Cos} \beta  + \operatorname{Sin} \beta \operatorname{Cos} \alpha  = \frac{3}{5} \times \left( {\frac{{ - 12}}{{13}}} \right) + \frac{5}{{13}} \times \frac{4}{5} = \frac{{ - 36}}{{65}} + \frac{{20}}{{65}} = \frac{{ - 16}}{{65}}  \cr 
  & \operatorname{Cos} (\alpha  - \beta ) = \operatorname{Cos} \alpha .\operatorname{Cos} \beta  + \operatorname{Sin} \alpha \operatorname{Sin} \beta  = \frac{4}{5} \times \left( {\frac{{ - 12}}{{13}}} \right) + \frac{3}{5} \times \frac{5}{{13}} = \frac{{ - 48}}{{65}} + \frac{{15}}{{65}} = \frac{{ - 33}}{{65}} \cr} $