Here:
4. Half Angle Formulas
It can be useful to wright:
sin(theta)=sin(2.(theta)/2)
...
Proof from the right side to the left side:
-numerator: sin^2(a)=sin^2(2.(a/2))=1-cos^2(2.(a/2))
Use now: (a^2-b^2)=(a-b)(a+b) and then the dubble angle formula.
-denominator: 1-cos(a)=1-cos(2.(a/2))
Use the dubble angle formula.
Dubble angle formule: cos(2a)=2cos^2(a)-1
let $\displaystyle t = \frac{\theta}{2}$
$\displaystyle 2\cos^2{t} = \frac{\sin^2(2t)}{1-\cos(2t)}$
on the RHS, use the following double angle identities ...
$\displaystyle \sin(2t) = 2\sin{t}\cos{t}$
$\displaystyle \cos(2t) = 1-2\sin^2{t}$
... you'll get to the LHS in a couple of steps.