Let's simplify the given expression:

$\left( \frac{x + iy}{\sqrt{x^2 + y^2}} \right)^2$

Step 1: Express in Polar Form

We recognize that $x + iy$ can be rewritten in polar form:

$x + iy = r e^{i\theta}$

where $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1} \left( \frac{y}{x} \right)$.

Thus, we rewrite the fraction:

$\frac{x + iy}{\sqrt{x^2 + y^2}} = \frac{r e^{i\theta}}{r} = e^{i\theta}$

Step 2: Square the Expression

$\left( e^{i\theta} \right)^2 = e^{i 2\theta}$

Step 3: Convert Back to Cartesian Form

Using Euler's formula:

$e^{i 2\theta} = \cos(2\theta) + i \sin(2\theta)$

Using the double-angle identities:

$\cos(2\theta) = \frac{x^2 - y^2}{x^2 + y^2}, \quad \sin(2\theta) = \frac{2xy}{x^2 + y^2}$

Thus, the simplified form of the given expression is:

$\frac{x^2 - y^2}{x^2 + y^2} + i \frac{2xy}{x^2 + y^2}$

Would you like a more detailed breakdown or an alternative approach?

Related Questions:

1. How do you convert complex numbers into polar form?
2. What are the Euler's and De Moivre’s theorems, and how do they help in complex number calculations?
3. What are the geometric interpretations of multiplying and squaring complex numbers?
4. How do you derive the double-angle identities for sine and cosine?
5. What is the significance of the unit circle in complex number calculations?

Tip: When working with complex fractions, converting them to polar form often simplifies exponentiation and division.