I tried the following approach.
Let . Then using de Moivre's theorem:
This is sum of GP. Using the sum of GP:
This simplifies to:
Now I changed the complex numbers on the RHS from Euler's form to cis form, then made the denominator real and finally took imaginary part of the expression. Is there a simpler way to solve this problem? My method is very complicated, thus error-prone.
In fact, I have already made an error which is why I have posted this note. I am getting the RHS as
i.e. without the term in the numerator. Where is the term coming from? There are only and in the equation. I can't seem to find out where I have gone wrong.