From: http://en.wikipedia.org/wiki/Tangent_half-angle_substitution
How did $\frac 1 {2\cos ^2 \frac x 2}$ become: $\frac {1+t^2} 2$?
From the substitution of $\cos x$, it should be similar to: $\frac 1 {\cos^2x} = \frac {t^2+1}{1-t^2}$ no?
I'm asking this because I'm trying to make this substitution for $\sin 2x, \cos 2x$, using $t=\tan 2x$ (and they turn out to be the same as the ones for $t=\tan \frac x 2$) but I can't find the $dx$.
The $t$ in these substitutions is $t=\tan(x/2)$, not $t=\tan x$ so that $$\sec^2(x/2)= \tan^2 (x/2)+1=t^2+1$$