The problem provided asks to solve for the expression $\frac{x}{y} + \frac{y}{x}$, given that:

$x = \cos\alpha + i\sin\alpha$ $y = \cos\beta + i\sin\beta$

Solution:

First, let's express $x$ and $y$ in exponential form using Euler's formula:

$x = e^{i\alpha}$ $y = e^{i\beta}$

Now, compute the ratio $\frac{x}{y}$:

$\frac{x}{y} = \frac{e^{i\alpha}}{e^{i\beta}} = e^{i(\alpha - \beta)}$

Next, compute its reciprocal $\frac{y}{x}$:

$\frac{y}{x} = e^{i(\beta - \alpha)}$

Now, let's calculate the required expression:

$\frac{x}{y} + \frac{y}{x} = e^{i(\alpha - \beta)} + e^{i(\beta - \alpha)}$

Notice that $e^{i(\alpha - \beta)} + e^{i(\beta - \alpha)}$ can be rewritten using the cosine identity $e^{i\theta} + e^{-i\theta} = 2\cos\theta$:

$\frac{x}{y} + \frac{y}{x} = 2\cos(\alpha - \beta)$

Final Answer:

$\frac{x}{y} + \frac{y}{x} = 2\cos(\alpha - \beta)$

Would you like any further details or explanations? Here are some related questions you might be interested in:

1. How does Euler's formula relate complex numbers to trigonometric functions?
2. What is the significance of the cosine identity $e^{i\theta} + e^{-i\theta} = 2\cos\theta$?
3. How can the result $2\cos(\alpha - \beta)$ be interpreted geometrically?
4. What other trigonometric identities can be derived using complex numbers?
5. How can this approach be applied to more complicated expressions involving complex numbers?

Tip: Understanding Euler's formula and how it relates to trigonometric functions is crucial when dealing with problems involving complex numbers in exponential form.