Animated Solution for Mathematics - Complex Numbers: If $z$ is a complex number of unit modulus and argument $\theta$, then $arg\left(\frac{1+z}{1+\bar{z}}\right)$ equals

Imagine you are standing on the Argand plane, the vast, two-dimensional playground where complex numbers live. We are given a complex number $z$ with a modulus of exactly one.

Geometrically, this is a beautiful constraint. It means our complex number $z$ is not just floating anywhere; it is anchored to the unit circle, a circle with a radius of one, centered right at the origin.

When we say the argument of $z$ is $\theta$, we are essentially drawing a vector from the origin to a point on this circle, making an angle of $\theta$ with the positive real axis. This vector is our $z$.

Now, consider its conjugate, $\bar{z}$. In the Argand plane, the conjugate is simply the reflection of $z$ across the real axis. If $z$ is at an angle of $\theta$, its reflection $\bar{z}$ will naturally be at an angle of $-\theta$.

Now, let us move from geometry to algebra. We need to simplify the expression:

To do this, we need a bridge between $z$ and its conjugate $\bar{z}$. Recall the fundamental property of complex numbers: a number multiplied by its conjugate always equals the square of its modulus, or $z\cdot \bar{z}=|z{|}^2$.

Since we know $|z|=1$, this equation becomes $z\cdot \bar{z}=1^2=1$. This is a powerful realization!

It tells us that for any complex number on the unit circle, $\bar{z}=\frac{1}{z}$. This substitution is a secret weapon in your JEE toolkit. Whenever you see $\bar{z}$ and $|z|=1$, think of $\frac{1}{z}$ immediately.

Now, let us tackle the expression itself: $\frac{1+z}{1+\bar{z}}$. We substitute $\bar{z}=\frac{1}{z}$ into the denominator.

The expression becomes:

This looks a bit messy, but let us focus on the denominator: $1+\frac{1}{z}$. By taking $z$ as the common denominator, we can rewrite this as $\frac{z+1}{z}$.

Now, substitute this back into our main fraction:

Notice the elegance here? The numerator is $(1+z)$, and the denominator is $\frac{z+1}{z}$. Since $(1+z)$ is the same as $(z+1)$, they cancel out perfectly!

We are left with $\frac{1}{1/z}$, which simplifies to just $z$.

We started with a complex-looking fraction, and through the power of the unit modulus property, we have reduced it to simply $z$. The question asks for the argument of this fraction.

Since the fraction is equal to $z$, its argument is simply the argument of $z$. And what is the argument of $z$? The problem gave it to us right at the start: it is $\theta$.

Thus, the argument of $\frac{1+z}{1+\bar{z}}$ is $\theta$.

It is a beautiful, clean result. This problem is a classic example of how, in JEE, the most intimidating expressions often collapse into something simple if you just look for the right geometric or algebraic property.