If a = cos 8 π 11 + i sin 8 π 11 , then Re a + a 2 + a 3 + a 4 + a 5 =

We have,

$a=\cos \left(\frac{8\pi }{11}\right)+i\sin \left(\frac{8\pi }{11}\right)$

Then,

$a^2={\left\{\cos \left(\frac{8\pi }{11}\right)+i\sin \left(\frac{8\pi }{11}\right)\right\}}^2$

      $=\cos \left(\frac{16\pi }{11}\right)+i\sin \left(\frac{16\pi }{11}\right)$

$a^3=\cos \left(\frac{24\pi }{11}\right)+i\sin \left(\frac{24\pi }{11}\right)$

$a^4=\cos \left(\frac{32\mathrm{\pi }}{11}\right)+i\sin \left(\frac{32\pi }{11}\right)$

$a^5=\cos \left(\frac{40\pi }{11}\right)+i\sin \left(\frac{40\pi }{11}\right)$

Now, 

$=a+a^2+a^3+a^4+a^5$

$=\cos \left(\frac{8\pi }{11}\right)+i\sin \left(\frac{8\pi }{11}\right)+\cos \left(\frac{16\pi }{11}\right)+i\sin \left(\frac{16\pi }{11}\right)+\cos \left(\frac{24\pi }{11}\right)+i\sin \left(\frac{24\pi }{11}\right)+\cos \left(\frac{32\pi }{11}\right)+i\sin \left(\frac{32\pi }{11}\right)+\cos \left(\frac{40\pi }{11}\right)+i\sin \left(\frac{40\pi }{11}\right)$

$=\left\{\cos \left(\frac{8\pi }{11}\right)+\cos \left(\frac{16\pi }{11}\right)+\cos \left(\frac{24\pi }{11}\right)+\cos \left(\frac{32\pi }{11}\right)\right\}+\cos \left(\frac{40\pi }{11}\right)+i\left\{\sin \left(\frac{8\pi }{11}\right)+\sin \left(\frac{16\pi }{11}\right)+\sin \left(\frac{24\pi }{11}\right)+\sin \left(\frac{32\pi }{11}\right)+\sin \left(\frac{40\pi }{11}\right)\right\}$

  $=\left\{\frac{\cos \left({\displaystyle \frac{8\mathrm{\pi }}{11}+4·\frac{8\mathrm{\pi }}{11}}\right)}{\sin \left({\displaystyle \frac{8\mathrm{\pi }}{11}}\right)}\times \sin \left(\frac{5\times 8\mathrm{\pi }}{11}\right)\right\}+\text{Imaginary part}$

$=\frac{\cos \left({\displaystyle \frac{40\mathrm{\pi }}{11}}\right)\sin \left({\displaystyle \frac{40\mathrm{\pi }}{11}}\right)}{\sin \left({\displaystyle \frac{8\mathrm{\pi }}{11}}\right)}+\text{Imaginary part}$        $=\left(-\frac{1}{2}\right)+\text{Imaginary part}$   

Therefore, $\text{Re}\left(a+a^2+a^3+a^4+a^5\right)=\frac{-1}{2}$