first of all not everything has to be simplified and second I think you better off staying in the first form --> cos(x) + sec^2(x) in order to find F''(x)
Okay the domain shoudl be straight forward
D={x is All Reals except when x = Pi/2 + K(Pi)} where K is an Odd Interger
A vertical Asymptote exists at Pi/2 + K(Pi) where K is an Odd Interger
F'(x)
just the addition rule, (tanx)' + (sinx)' = sec^2x + cosx
so simpleify to cos
you end up with 1+cos^3x/cos^2, I must be missing an identity to simplify this, any ideas?