Help with complex summation further maths a levels

Please help with 9231 MJ 14 13 Q5

It would help to post a picture of the question as well as what youve tried so far / what youre stuck with.

For some reason it won't allow me to post a picture

Please help with 9231 MJ 14 13 Q5

For some reason it won't allow me to post a picture

I think you are referring to this question.
If yes, the following should help.
z + z² + z³ + … zⁿ is a geometric series.

To prove the sum of the series of the cosine terms, we can use de Moivre’s theorem and then consider the result of z + z² + z³ + … zⁿ and use the real part.

If the question is incorrect, please state the following
Year:
Jun or Nov:
Paper version:
Question:

Hint: z^n + 1/z^n = 2cos(nθ)

Just for info/fun, you could try and do the question using either geometry/trig or using trig identity/telescoping(differences). Both are a bit less algebra than using a complex geometric sequence which is the way the question asks for.

If you aggressively look for conjugates the GM method comes out pretty quickly as well (obviously all the methods are "pretty much the same" underneath). I somewhat dislike the fact they've gone for a sin(A)cos(B)/sin(C) form of final answer as opposed to (cos(X)-cos(Y))/sin(C) - it doesn't feel to me like an obviously simpler answer and it obfuscates what's going on somewhat.

I'm not sure doing z^n+1/z^n helps much here, particularly given you've already done z+z^2+...+z^n for the 1st part of the question. Just take the real part of that sum (conjugates are your friend).