Question

Sec^2(u/2)=(2sec(u))/(sec(u)+1)

Answer 1

It would be helpful to know what you need with this identity. Prove it? One wonders.
If so, you need to start on another identity.
Remember cos(2x)=cos^2(x) - sin^2(x)=cos^2(x)-(1-cos^2(x))=2cos^2(x)-1 or
cos^2(x)=1/2 (1+cos(2x))
now replace 2x=u and you have
cos^2(u/2)=1/2 (1+cos(u))
and
sec^2(u/2)=2/(1+1/sec(u))
now multiply the right side by secu/secu
sec^2(u/2)=2sec(u)/(sec(u)+1)
viola.