dx/dy=[sec^2(y/2)]/2 is good (there's NO dy/dx at the end btw). But forget about the 2/[cos^2(y/2)] = 2/1-sin^2(y/2) business .....

dx/dy=[sec^2(y/2)]/2 therefore dy/dx = 2/[sec^2(y/2)].

Now remember the standard identity tan^2(y/2) + 1 = sec^2(y/2).

But x=tan(y/2).

Therefore dy/dx = ....