(cos@+cos#)/(sin@+sin#)=(sin@+sin#)/(cos@+cos#)(cos@+cos#)/(sin@+sin#)=(sin@+sin#)/(cos@+cos#)

The equation you provided is:

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

To solve or simplify this, let's analyze the equation step by step.

### Step 1: Cross-Multiply to Eliminate the Fractions
Cross-multiplying the fractions gives:

\[
(\cos \alpha + \cos \beta)^2 = (\sin \alpha + \sin \beta)^2
\]

### Step 2: Expand Both Sides
Expanding both sides using the identity \((a + b)^2 = a^2 + 2ab + b^2\):

\[
(\cos \alpha)^2 + 2 \cos \alpha \cos \beta + (\cos \beta)^2 = (\sin \alpha)^2 + 2 \sin \alpha \sin \beta + (\sin \beta)^2
\]

### Step 3: Use the Pythagorean Identity
We know that \(\cos^2 \theta + \sin^2 \theta = 1\), so:

\[
1 + 2 \cos \alpha \cos \beta = 1 + 2 \sin \alpha \sin \beta
\]

### Step 4: Simplify the Equation
Subtract 1 from both sides:

\[
2 \cos \alpha \cos \beta = 2 \sin \alpha \sin \beta
\]

### Step 5: Divide by 2
Dividing both sides by 2:

\[
\cos \alpha \cos \beta = \sin \alpha \sin \beta
\]

### Step 6: Use the Product-to-Sum Identity
The product-to-sum identity states:

\[
\cos \alpha \cos \beta = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)]
\]
\[
\sin \alpha \sin \beta = \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)]
\]

Thus, we have:

\[
\cos(\alpha - \beta) + \cos(\alpha + \beta) = \cos(\alpha - \beta) - \cos(\alpha + \beta)
\]

### Step 7: Solve the Equation
Subtract \(\cos(\alpha - \beta)\) from both sides:

\[
2 \cos(\alpha + \beta) = 0
\]

This implies:

\[
\cos(\alpha + \beta) = 0
\]

This indicates that \(\alpha + \beta\) is an odd multiple of \(\frac{\pi}{2}\).

So, the equation holds true when \(\alpha + \beta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots\).

### Conclusion
The given equation is valid under the condition that \(\alpha + \beta\) is an odd multiple of \(\frac{\pi}{2}\).

---

Would you like more details or have any questions? 

Here are some related questions for further exploration:

1. What are the general solutions for \(\cos(\theta) = 0\)?
2. How can you derive the product-to-sum identities?
3. What is the significance of the Pythagorean identity in trigonometry?
4. Can the original equation be simplified under other conditions?
5. How do trigonometric identities simplify complex trigonometric expressions?
6. What are the implications if \(\alpha + \beta\) equals an even multiple of \(\frac{\pi}{2}\)?
7. How do you prove the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\)?
8. What are some applications of trigonometric identities in real-world problems?

**Tip:** When solving trigonometric equations, always consider the possibility of using identities to simplify the problem.

Explore the solution to the trigonometric equation (cos@+cos#)/(sin@+sin#)=(sin@+sin#)/(cos@+cos#). Understand how to apply product-to-sum identities and the Pythagorean identity in solving trigonometric equations. Suitable for advanced high school students and anyone interested in advanced trigonometry problems.

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