The expression `(sinalpha+sinbeta)/(cosalpha+cosbeta)` is equal to

Options

A. `tan((alpha+beta)/(2))`
B. `cot((alpha+beta)/(2))`
C. `sin((alpha+beta)/(2))`
D. `cos((alpha+beta)/(2))`

Explanation

To solve the expression \(\frac{\sin \alpha + \sin \beta}{\cos \alpha + \cos \beta}\), we can use trigonometric identities to simplify it step by step.

### Step 1: Write the expression
Let:
\[ x = \frac{\sin \alpha + \sin \beta}{\cos \alpha + \cos \beta} \]

### Step 2: Use the sum-to-product identities
We can apply the sum-to-product identities for sine and cosine:
\[ \sin \alpha + \sin \beta = 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \]
\[ \cos \alpha + \cos \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \]

### Step 3: Substitute the identities into the expression
Substituting these identities into our expression for \(x\):
\[ x = \frac{2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)}{2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)} \]

### Step 4: Simplify the expression
The \(2\) and \(\cos\left(\frac{\alpha - \beta}{2}\right)\) terms cancel out:
\[ x = \frac{\sin\left(\frac{\alpha + \beta}{2}\right)}{\cos\left(\frac{\alpha + \beta}{2}\right)} \]

### Step 5: Recognize the tangent function
This simplifies to:
\[ x = \tan\left(\frac{\alpha + \beta}{2}\right) \]

### Conclusion
Thus, we find that:
\[ \frac{\sin \alpha + \sin \beta}{\cos \alpha + \cos \beta} = \tan\left(\frac{\alpha + \beta}{2}\right) \]