Answer
$cos~(2~tan^{-1}~(-2)) = -\frac{3}{5}$
Work Step by Step
Let $~~\theta = tan^{-1}(-2)$
Then: $~~tan~\theta = -2 = \frac{-2}{1}$
Then: $~~sin~\theta = \frac{-2}{\sqrt{(-2)^2+(1)^2}} = -\frac{2}{\sqrt{5}}$
We need to find $cos~2\theta$:
$cos~2\theta = 1-2~sin^2~\theta$
$cos~2\theta = 1-2~(-\frac{2}{\sqrt{5}})^2$
$cos~2\theta = 1-\frac{8}{5}$
$cos~2\theta = -\frac{3}{5}$
Therefore, $~~cos~(2~tan^{-1}~(-2)) = -\frac{3}{5}$