Prove this formula $1+\cos\theta+\cos2\theta+...+\cos n\theta=\frac{1}{2}+\frac{\sin(n+\frac{1}{2})\theta}{2\sin\frac{\theta}{2}}$

Question

This is homework but I’m really stuck. The question is to prove a fromula which states:

$$1+\cos\theta+\cos2\theta+...+\cos n\theta=\frac{1}{2}+\frac{\sin(n+\frac{1}{2})\theta}{2\sin\frac{\theta}{2}}$$

I want to solve it using complex numbers So I came to this (I missed Re in last one) Can you guys please help me finish this ?

Or this: https://math.stackexchange.com/q/3404544/42969 or this: https://math.stackexchange.com/q/17966/42969. — Martin R, Oct 04 '20 at 07:21
@rtybase Thank you for your consideration,I already found an answer.but thank you for your time. I checked it .It was also helpful. — Negar Rezaei Nejad, Oct 04 '20 at 08:51

kuspia · Answer 1 · 2020-10-04T10:06:21.633

1

You can refer this :

This way you can easily do to prove Lagrange’s trigonometric identity:

edited Oct 04 '20 at 10:06

answered Oct 04 '20 at 06:53

kuspia

1

$2Sin(-\frac{\theta}{2})=-2Sin(\frac{\theta}{2})$.In your answer ,seventh line , what happened to the - ? – Negar Rezaei Nejad Oct 04 '20 at 07:34
Cause we know that $2i Sinx=e^{ix}-e^{-ix}$ – Negar Rezaei Nejad Oct 04 '20 at 07:38
@Negar have u understood it completely or still not ? – kuspia Oct 04 '20 at 08:45
This looks like a screenshot taken from http://faculty.bard.edu/belk/math362/Homework1Solutions.pdf. Please have a look at https://math.stackexchange.com/help/referencing about to reference material written by others, with proper attribution. – Martin R Oct 04 '20 at 10:03

1 Answers1