`(cos alpha + cos beta)/( sin alpha - sin beta) + (sin alpha + sin beta)/( cos alpha - cos beta )` =

Options

A. 1
B. 2
C. `sqrt(2)`
D. 0

Answer

To solve the expression

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

we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((\sin \alpha - \sin \beta)(\cos \alpha - \cos \beta)\). We will rewrite the expression using this common denominator.

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

### Step 2: Expand the numerators
Now, we will expand the numerators of both terms.

1. For the first term:
\[ (\cos \alpha + \cos \beta)(\cos \alpha - \cos \beta) = \cos^2 \alpha - \cos^2 \beta \]

2. For the second term:
\[ (\sin \alpha + \sin \beta)(\sin \alpha - \sin \beta) = \sin^2 \alpha - \sin^2 \beta \]

Combining these, we have:
\[ \cos^2 \alpha - \cos^2 \beta + \sin^2 \alpha - \sin^2 \beta \]

### Step 3: Use the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can rewrite the expression:
\[ (\cos^2 \alpha + \sin^2 \alpha) - (\cos^2 \beta + \sin^2 \beta) = 1 - 1 = 0 \]

### Step 4: Final expression
Thus, the numerator simplifies to 0. Therefore, the entire expression simplifies to:
\[ \frac{0}{(\sin \alpha - \sin \beta)(\cos \alpha - \cos \beta)} = 0 \]

### Conclusion
The final answer is:
\[ 0 \]

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