The question is: Show that $\displaystyle (tan(x/2))' = 1/(1+cos(x))$.
$\displaystyle tan(x/2)' = 1/(2(cos (x/2)^2)$, but I have no idea where to go from here.
You need to know about half angle formula. Look at this:
List of trigonometric identities - Wikipedia, the free encyclopedia