sums of series involving complex numbers

Question

I dont know how to use latex any good rescources to quickly learn from scratch, hence why I have photographed the question.

The question is as follows

question

question final part of working

The solution is the book to me does not seem very elegant and perhaps long winded so is there a better way to derive the result. For example without changing from the exponential to the modulus argument form.

Maybe use double angle formulae might be better but I am unsure.

Thanks,

Daniel

(PS I dont have enough rep to include the image)

Answer 1

I think the solution that is done in the book is good enough,it is not long-winded for me as you are just using the formula for Geometric series and $$\exp(ia)=\cos a+i\sin a$$ and the rest is just basic manipulations.