There seems to be a dropped a minus sign in the last line of the answer to the first part.

I believe Penrose got a minus sign switched around in the exercise statement. It ought to read:

"Show that replacing

by

or by

corresponds, respectively, to replacing

by

or by

."

(Note the change of position of the minus).

That would correspond correctly to the first part of the posted solution after the missing minus sign is restored, and also to the second part of the posted solution.

Note also that an easier way of doing the second part is to do it exactly the same way as the first part, but using the other Fourier formula near the top of the page (instead of the inverse Fourier formula near the bottom).