Answer
$$ - \frac{{16}}{{65}}$$
Work Step by Step
$$\eqalign{
& \cos \left( {{{\sin }^{ - 1}}\frac{3}{5} + {{\cos }^{ - 1}}\frac{5}{{13}}} \right) \cr
& {\text{Let }}\theta = {\sin ^{ - 1}}\frac{3}{5}{\text{ and }}\beta = {\cos ^{ - 1}}\frac{5}{{13}} \cr
& \cr
& {\text{Therefore,}} \cr
& \theta = {\sin ^{ - 1}}\frac{3}{5} \to \sin \theta = \frac{3}{5} = \frac{{{\text{opposite side}}}}{{{\text{hypotenuse}}}} \cr
& {\text{adjacent side}} = \sqrt {{5^2} - {3^2}} = 4 \cr
& and \cr
& \beta = {\cos ^{ - 1}}\frac{5}{{13}} \to \cos \theta = \frac{5}{{13}} = \frac{{{\text{adjacent side}}}}{{{\text{hypotenuse}}}} \cr
& {\text{opposite side}} = \sqrt {{{13}^2} - {5^2}} = 12 \cr
& \cr
& \cos \left( {{{\sin }^{ - 1}}\frac{3}{5} + {{\cos }^{ - 1}}\frac{5}{{13}}} \right) = \cos \left( {\theta + \beta } \right) \cr
& {\text{Use the cosine of a sum}} \cr
& = \cos \theta \cos \beta - \sin \theta \sin \beta \cr
& = \left( {\frac{4}{5}} \right)\left( {\frac{5}{{13}}} \right) - \left( {\frac{3}{5}} \right)\left( {\frac{{12}}{{13}}} \right) \cr
& {\text{Simplify}} \cr
& = - \frac{{16}}{{65}} \cr} $$