trigonometry - solve$, ;cosleft(,2arctanleft(1/2right),right),$

How would I go on about solving:
$\,\quad\cos\Big(\,2\cdot \arctan\big(\frac{1}{2}\big)\Big)$ ?

My attempt was to find $\,v\,$ in $\,\frac{\sin\left(v\right)}{\cos\left(v\right)} = \frac{1}{2}\,$ and substitute that with $\,\arctan\left(\frac{1}{2}\right),\,$ but I end up with $\,\sin\left(v\right) = \sqrt{\frac{1}{5}}\,$ which I cant solve.

Is there any other way to approach the problem?