Use the Cosine Double Angle Formula again, (NOT SINE DOUBLE ANGLE) to help derive Cosine 1/2 Angle.

$\displaystyle \text{cos}\ 2A=1-2{{\sin }^{2}}x$

Substitute A for 1/2 x
$\displaystyle \text{cos}\ 2\left( {\frac{1}{2}x} \right)=1-2{{\sin }^{2}}\left( {\frac{1}{2}x} \right)$

Simplify the equation (*Note: You can multiply cos 2(1/2x) to get cos x.)
$\displaystyle \text{cos}\ x=1-2{{\sin }^{2}}\left( {\frac{1}{2}x} \right)$
$\displaystyle 2{{\sin }^{2}}\left( {\frac{1}{2}x} \right)=1-\text{cos}\ x$
$\displaystyle {{\sin }^{2}}\left( {\frac{1}{2}x} \right)=\frac{{1-\text{cos}\ x}}{2}$
$\displaystyle \sin \left( {\frac{1}{2}x} \right)=\sqrt{{\frac{{1-\text{cos}\ x}}{2}}}$

In the previous post I show you how to derive cosine 1/2 Angle. You will notice two similarities between cos 1/2 x and sin 1/2 x. Firstly, both will derive to almost the same expression, with the only exception being one is plus and the other minus. Secondly, You have to use Cosine double angle to derive both cosine 1/2 angle and sine 1/2 angle. It seems a bit counter-intuitive cause we would expect to use Sine double angle to derive sine 1/2 angle. You can give it a try, but you will never get the answer.