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 ?

Answer 1

You can refer this :

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

http://faculty.bard.edu/belk/math362/Homework1Solutions.pdf original answer