Attached my solution.

Just a couple of remarks:

(1) On the line below (*), the term after the rightmost "=" should be . You incorrectly left out the brackets.

(2) Part (b) is probably simpler if you start from the original expression for , rather than from (*).

(1) I attached here the corrected solution, adding the missing brackets and correcting a couple of spelling errors.

(2) Probably the approach you mention would be slightly simpler; however I found esthetically better not to use the explicit expression for *F, even if it is provided by Penrose at page 443, because I never proved it, even if it would be straightforward to do.