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  4. The value of ((1+ sin (2 π/9)+i cos (2 π/9)/1+ sin (2 π)9-i cos (2 π)9)3 is

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Q. The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3$ is

JEE MainJEE Main 2023Complex Numbers and Quadratic Equations

Solution:

Let $\sin \frac{2 \pi}{9}+ i \cos \frac{2 \pi}{9}= z$
$ \left(\frac{1+ z }{1+\bar{z}}\right)^3=\left(\frac{1+ z }{1+\frac{1}{ z }}\right)^3= z ^3 $
$ \Rightarrow\left( i \left(\cos \frac{2 \pi}{9}- i \sin \frac{2 \pi}{9}\right)\right)^3$
$ =-i\left(\cos \frac{2 \pi}{3}- i \sin \frac{2 \pi}{3}\right)=-i\left(\frac{-1}{2}- i \frac{\sqrt{3}}{2}\right) $
$ \Rightarrow \frac{-1}{2}(\sqrt{3}-i) .$