Find $\sin(\frac{x}{2})$, given $\tan(x) = 2$, with $0 < x < \frac{\pi}{2}$.

Question

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

Answer 1

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

Answer 2

$$\dfrac1{\cos x}=\sec x=+\sqrt{1+\tan^2x}$$

Now $\cos2y=1-2\sin^2y$

Here $\sin\dfrac x2>0$ right?