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 enter image description here (I missed Re in last one) Can you guys please help me finish this ?

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    $\begingroup$ Multiply top and bottom with $(1-\cos\theta)+i\sin\theta$. $\endgroup$ Commented Oct 4, 2020 at 6:44
  • $\begingroup$ Or this: math.stackexchange.com/q/3404544/42969 or this: math.stackexchange.com/q/17966/42969. $\endgroup$
    – Martin R
    Commented Oct 4, 2020 at 7:21
  • $\begingroup$ @MartinR thank you for your consideration. It did <3 $\endgroup$ Commented Oct 4, 2020 at 7:47
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    $\begingroup$ @rtybase Thank you for your consideration,I already found an answer.but thank you for your time. I checked it .It was also helpful. $\endgroup$ Commented Oct 4, 2020 at 8:51

1 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

click here

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    $\begingroup$ $2Sin(-\frac{\theta}{2})=-2Sin(\frac{\theta}{2})$.In your answer ,seventh line , what happened to the - ? $\endgroup$ Commented Oct 4, 2020 at 7:34
  • $\begingroup$ Cause we know that $2i Sinx=e^{ix}-e^{-ix}$ $\endgroup$ Commented Oct 4, 2020 at 7:38
  • $\begingroup$ @Negar have u understood it completely or still not ? $\endgroup$
    – kuspia
    Commented Oct 4, 2020 at 8:45
  • $\begingroup$ This looks like a screenshot taken from faculty.bard.edu/belk/math362/Homework1Solutions.pdf. Please have a look at math.stackexchange.com/help/referencing about to reference material written by others, with proper attribution. $\endgroup$
    – Martin R
    Commented Oct 4, 2020 at 10:03

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