Prove sum of $\cos(\pi/11)+\cos(3\pi/11)+...+\cos(9\pi/11)=1/2$ using Euler's formula Prove that $$\cos(\pi/11)+\cos(3\pi/11)+\cos(5\pi/11)+\cos(7\pi/11)+\cos(9\pi/11)=1/2$$
using Euler's formula.
Everything I tried has failed so far.
Here is one thing I tried, but obviously didn't work.
$$\Re e \{e^{\frac{\pi}{11}i}(1+e^{\frac{2\pi}{11}i}+e^{\frac{4\pi}{11}i}+e^{\frac{6\pi}{11}i}+e^{\frac{8\pi}{11}i}) \}=\frac{1}{2}$$
$$\Re e \{e^{\frac{\pi}{11}i}(1+\sqrt[11]{e^{2\pi i}}+\sqrt[11]{e^{4\pi i}}+\sqrt[11]{e^{6\pi i}}+\sqrt[11]{e^{8\pi i}}) \}=\frac{1}{2}$$
$$\Re e \{e^{\frac{\pi}{11}i}(1+\sqrt[11]{1}+\sqrt[11]{1}+\sqrt[11]{1}+\sqrt[11]{1}) \}=\frac{1}{2}$$
$$\Re e \{5e^{\frac{\pi}{11}i} \}=\frac{1}{2}$$
$$5\cos(\frac{\pi}{11})=\frac{1}{2}$$
Which isn't true :D
Thanks in advance
 A: Put $$S = \cos(π/11)+\cos(3π/11)+\cos(5π/11)+\cos(7π/11)+\cos(9π/11)$$
Then $$S = \cos(-π/11)+\cos(-3π/11)+\cos(-5π/11)+\cos(-7π/11)+\cos(-9π/11)$$ (because cos is even)
and of course $$ -1 = \cos (-11\pi/11)$$
Sum these all up and you get $$2S-1$$ as the sum of the real parts of the eleven eleventh-roots of unity, which is 0, and therefore $$S = 1/2$$
A: HINT:
First of all, $$e^x=1\iff x=2n\pi$$ where $n$ is some integer
$\sum_{r=0}^5\cos\frac{(2r+1)\pi}{11}$
=Re[$\sum_{r=0}^5\left(e^{\frac{\pi i}{11}}\right)^{2r+1}$]
Now  $\sum_{r=0}^5\left(e^{\frac{\pi i}{11}}\right)^{2r+1}$ is a Geometric Series 
A: Sum of cosines of $n$ angles in Arithmetic Progression ( starting at $ \alpha$, common angle difference $\beta$ ) has a ready formula, with simple trig derivation/simplification:
$$ \cos \alpha + \cos ( \alpha + \beta )+  \cos ( \alpha + 2\beta) +  \cos ( \alpha + (n-1) \beta   = \dfrac { \sin (n \beta/2)} { \sin (\beta/2) }\cdot \cos ( \alpha + {(n+1)\beta/2} ).   $$
The same is likewise valid for sines, the quotient does not change.
$$ \sin \alpha + \sin ( \alpha + \beta )+  \sin ( \alpha + 2\beta) +  \sin ( \alpha + (n-1) \beta   = \dfrac { \sin (n \beta/2)} { \sin (\beta/2) }\cdot \sin ( \alpha + {(n+1)\beta/2} ).  $$
EDIT1:
Multiply second equation with $ i$ and add to the first one. Use  Euler's relation in a geometric series with constant ratio terms, common difference is in the exponent. 
