Find $\sin(\frac{x}{2})$, given $\tan(x) = 2$, with $0 < x < \frac{\pi}{2}$. Find $\sin(\frac{x}{2})$, given $\tan(x) = 2$, with $0 < x < \frac{\pi}{2}$. Which half-identity formular should I use and why?

$\tan^2 x + 1 =\sec^2 x\\\ \sec x = \sqrt 5\\\ \cos x = \frac {1}{\sqrt 5}\\\ \sin \frac{x}{2} = \sqrt {\frac {1-\cos x}{2}}$