Explanation

(b)
$\left\lbrack 1 + \cos{\frac{\pi}{5} + i\sin\frac{\pi}{5}} \right\rbrack^{- 1} = \frac{1}{2\cos^{2}{\frac{\pi}{10} + 2\ i\sin{\frac{\pi}{10}\cos\frac{\pi}{10}}}}$
$= \frac{1}{2\cos{\frac{\pi}{10}\left( \cos{\frac{\pi}{10} + i\sin\frac{\pi}{10}} \right)}} \times \frac{\cos{\frac{\pi}{10} - i\sin\frac{\pi}{10}}}{\left( \cos{\frac{\pi}{10} - i\sin\frac{\pi}{10}} \right)}$
$= \frac{\cos{\frac{\pi}{10} - i\sin\frac{\pi}{10}}}{2\cos\frac{\pi}{10}}$
$\therefore$ Real part is $\frac{1}{2}$