The expression you've provided is:

\[
\frac{\cos \alpha + \cos \beta}{\sin \alpha + \sin \beta}
\]

To simplify or analyze this expression, we can use trigonometric identities and properties. Let's explore this step by step.

### Step 1: Use Sum-to-Product Identities
The sum-to-product identities can help simplify expressions like \(\cos \alpha + \cos \beta\) and \(\sin \alpha + \sin \beta\).

The identities are:
\[
\cos \alpha + \cos \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)
\]
\[
\sin \alpha + \sin \beta = 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)
\]

### Step 2: Apply the Identities
Substituting these into the given expression:

\[
\frac{\cos \alpha + \cos \beta}{\sin \alpha + \sin \beta} = \frac{2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)}{2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)}
\]

### Step 3: Simplify the Expression
The \(\cos\left(\frac{\alpha - \beta}{2}\right)\) terms in the numerator and denominator cancel out (assuming \(\cos\left(\frac{\alpha - \beta}{2}\right) \neq 0\)):

\[
\frac{\cos\left(\frac{\alpha + \beta}{2}\right)}{\sin\left(\frac{\alpha + \beta}{2}\right)}
\]

This simplifies to:

\[
\cot\left(\frac{\alpha + \beta}{2}\right)
\]

### Final Answer
The expression simplifies to:

\[
\frac{\cos \alpha + \cos \beta}{\sin \alpha + \sin \beta} = \cot\left(\frac{\alpha + \beta}{2}\right)
\]

This is the cotangent of the average of the angles \(\alpha\) and \(\beta\).

---

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What are other sum-to-product identities and how are they derived?
2. How can you interpret the cotangent function geometrically?
3. What happens if \(\cos\left(\frac{\alpha - \beta}{2}\right) = 0\) in this problem?
4. Can you derive the expression for \(\frac{\sin \alpha + \sin \beta}{\cos \alpha + \cos \beta}\)?
5. What is the significance of simplifying trigonometric expressions in calculus?
6. How do product-to-sum identities relate to Fourier series?
7. What are some practical applications of trigonometric identities in engineering?
8. How can we use trigonometric identities to solve trigonometric equations?

**Tip:** When working with trigonometric expressions, recognizing patterns and applying identities can greatly simplify the problem.

Explore how to simplify the trigonometric expression (cos@ + cos#) / (sin@ + sin#) using sum-to-product identities. Learn about the cotangent function and its application in trigonometric simplification. Suitable for advanced high school students and anyone interested in trigonometry.

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