Answer
$\sin 2\theta=\frac{2\sqrt 2}{3}$ and $\cos 2\theta=-\frac{1}{3}$
Work Step by Step
$\sec^{2}\theta=1+tan^{2}\theta=1+2=3$
$\implies \sec\theta= \sqrt 3$ ($\sec \theta$ is positive as $0\leq\theta\lt\frac{\pi}{2}$)
$\cos\theta=\frac{1}{\sec\theta}=\frac{1}{\sqrt 3}$
$\sin^{2}\theta=1-\cos^{2}\theta=1-\frac{1}{3}=\frac{2}{3}$
$\implies \sin \theta=\frac{\sqrt 2}{\sqrt 3}$ ($\sin \theta$ is positive as $0\leq\theta\lt\frac{\pi}{2}$)
Now, $\sin 2\theta=2\sin\theta\cos\theta=2\times\frac{\sqrt 2}{\sqrt 3}\times\frac{1}{\sqrt 3}=\frac{2\sqrt 2}{3}$
$\cos 2\theta= \cos^{2}\theta-\sin^{2}\theta=\frac{1}{3}-\frac{2}{3}=-\frac{1}{3}$
