Answer
$$ - \frac{7}{{25}}$$
Work Step by Step
$$\eqalign{
& \cos \left( {2\arctan \frac{4}{3}} \right) \cr
& {\text{Let }}\theta = \arctan \frac{4}{3}{\text{, thus}} \cr
& \tan \theta = \frac{4}{3} \cr
& {\text{Recall that }}\tan \theta = \frac{{{\text{opposite side}}}}{{{\text{adjacent side}}}} \cr
& {\text{opposite side }} = 4 \cr
& {\text{adjacent side}} = 3 \cr
& {\text{hypotenuse}} = 5 \cr
& \cr
& {\text{We have that }}\cos \left( {2\arctan \frac{4}{3}} \right) = \cos \left( {2\theta } \right) \cr
& {\text{Use the identity }}\cos 2\theta = 1 - 2{\sin ^2}\theta \cr
& \cos \left( {2\arctan \frac{4}{3}} \right) = 1 - 2{\left( {\frac{4}{5}} \right)^2} \cr
& \cos \left( {2\arctan \frac{4}{3}} \right) = - \frac{7}{{25}} \cr} $$