The value of ((1 + sin(2π)/(9) + icos(2π)/(9))/(1 + sin(2π)/(9) -...

Given,

${\left(\dfrac{1 + sin\dfrac{2\pi}{9} + icos\dfrac{2\pi}{9}}{1 + sin\dfrac{2\pi}{9} - icos\dfrac{2\pi}{9}}\right)}^{3}$

Now let $z = sin\dfrac{2\pi}{9} + icos\dfrac{2\pi}{9}$,

So, $\overset{¯}{z} = sin\dfrac{2\pi}{9} - icos\dfrac{2\pi}{9} = \dfrac{1}{z}$

So, ${\left(\dfrac{1 + sin\dfrac{2\pi}{9} + icos\dfrac{2\pi}{9}}{1 + sin\dfrac{2\pi}{9} - icos\dfrac{2\pi}{9}}\right)}^{3}$

$= {\left(\dfrac{1 + z}{1 + \overset{¯}{z}}\right)}^{3}$

$= {\left(\dfrac{1 + z}{1 + \dfrac{1}{z}}\right)}^{3}$

$= {z}^{3}{\left(\dfrac{1 + z}{1 + z}\right)}^{3}$

$= {z}^{3}$

$= {\left(sin\dfrac{2\pi}{9} + icos\dfrac{2\pi}{9}\right)}^{3}$

$= {i}^{3}{\left(cos\dfrac{2\pi}{9} - isin\dfrac{2\pi}{9}\right)}^{3}$

$= - i\left(\cos \left(3 \times \dfrac{2\pi}{9}\right) - isin\left(3 \times \dfrac{2\pi}{9}\right)\right)$

$= - i\left(cos\dfrac{2\pi}{3} - isin\dfrac{2\pi}{3}\right)$

$= - i\left(\dfrac{- 1}{2} - i\dfrac{\sqrt{3}}{2}\right)$

$= - \dfrac{1}{2}\left(\sqrt{3} - i\right)$