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