Like the five mark question on the EXT2 paper that you was the only question you could not do and as soon as you walk out of the exam room, the solution is as clear as a bell in your mind...
Yes I love mental blocks.
"just one Q, with this question, did you know to expand the binomials (evetnually leading to a quadratic in x^2) or just tried it in an attempt to answer the Q?"
I think what he did was looked at the question, noting that the answer was in terms of tan^2 [on the RHS] and there was a square root [on the LHS], recognising the quadratic formula form.
As you said, it is easy to see that the expression w.r.t z cancels to form a quartic [z^4] and I would have concluded that the quadratic solution [from the quadratic formula] to the quartic must be in terms of tan^2, meaning that the polynomial must be expanded.
You have to be able to intuitively see the link between the "tan^2=" result of the question and the "z^4" result of the expansion of the previous part. I had to read it a couple of times. lol.