Hi overlords.

I have a pretty grand request if someone has the time. In my revision for an upcoming exam there is a fairly advanced (for intro level LA) question regarding eigenvectors. Now, I'm very familiar with what eigenvectors are and how to obtain them, however because of my poor math background I am really unfamiliar with factorising trig functions and have literally no idea how to operate with imaginary/complex numbers. Could someone possibly give a 10 seconds lesson on how to approach this question after the determinant has been found? Everything above the blue line I understand, everything else I need some help.

I understand this is quite a task, but I don't have the time to fully learn all of the above, as well as revise for everything else in time for exams, so I figured I'd ask here as a last resort. No feelings hurt if no takers