Answer
$\dfrac{4}{3}$
Work Step by Step
We have: $\sin \theta=\dfrac{2}{\sqrt 5}$ and $\cos \theta=-\dfrac{1}{\sqrt 5}$
Since, $\tan (\theta)=\dfrac{\sin \theta}{\cos \theta} \implies \tan \theta=\dfrac{2/\sqrt 5}{-1/\sqrt 5}=-2$
We plug these values into the following identity:
$\tan (2 \ \theta)=\dfrac{2 \tan \theta}{ 1-\tan^2 \theta} = \dfrac{(2)(-2) }{ 1- (-2)^2}=\dfrac{4}{3}$
