Animated Solution for Mathematics - Complex Numbers: 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

When you first look at the expression

your instinct might be to panic. You might think about expanding or rationalizing the denominator.

In JEE Advanced, if a problem looks like a calculation nightmare, it is almost certainly a test of your ability to recognize symmetry. Stop, breathe, and look for the underlying structure.

Let us simplify our life by setting $\theta =\frac{2\pi }{9}$. Our expression now involves $z=\sin \theta +i\cos \theta$.

Notice the structure: the numerator is $1+z$, and the denominator is $1+\sin \theta -i\cos \theta$. That denominator is the complex conjugate of $z$, which we denote as $\bar{z}$.

So, the entire expression is simply:

This is the moment where the problem shifts from a calculation task to a conceptual one.

Now, look at $z=\sin \theta +i\cos \theta$. Its modulus is:

This is a unit circle complex number. A fundamental property of any complex number with a modulus of $1$ is that $z\bar{z}=|z{|}^2=1$, which implies $\bar{z}=\frac{1}{z}$.

By substituting $\bar{z}=\frac{1}{z}$ into our fraction, we get:

Look at the denominator: $1+\frac{1}{z}$ is just $\frac{z+1}{z}$. When you divide $(1+z)$ by $\frac{z+1}{z}$, the $(1+z)$ terms cancel out beautifully, leaving you with just $z$. The entire terrifying fraction has collapsed into a single variable.

We are left with $z^3$. However, we must be careful; our $z$ was $\sin \theta +i\cos \theta$.

To use De Moivre's Theorem, we need the standard form $\cos \varphi +i\sin \varphi$. We use complementary angles: $\sin \theta =\cos (\frac{\pi }{2}-\theta )$ and $\cos \theta =\sin (\frac{\pi }{2}-\theta )$.

With $\theta =\frac{2\pi }{9}$, the argument $\varphi$ is:

So, $z=e^{i\frac{5\pi }{18}}$. Cubing this gives:

Finally, we convert back to trigonometric form:

Factoring out $-\frac{1}{2}$, we arrive at the final answer:

You see? No brute force, just elegance. Keep this mindset, and you will conquer any problem.