`sin alpha+sin beta=(1)/(4)` and `cos alpha+cos beta=(1)/(3)`
The value of `cos(alpha+beta)` is

Options

A. `(12)/(25)`
B. `(7)/(25)`
C. `(12)/(13)`
D. none of these.

Explanation

To find the value of \( \cos(\alpha + \beta) \) given the equations \( \sin \alpha + \sin \beta = \frac{1}{4} \) and \( \cos \alpha + \cos \beta = \frac{1}{3} \), we can follow these steps:

### Step 1: Use the sum-to-product identities
We can express \( \sin \alpha + \sin \beta \) and \( \cos \alpha + \cos \beta \) using the sum-to-product identities:
\[ \sin \alpha + \sin \beta = 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \]
\[ \cos \alpha + \cos \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) \]

### Step 2: Set up the equations
From the given equations, we can set up:
\[ 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) = \frac{1}{4} \quad \text{(1)} \]
\[ 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) = \frac{1}{3} \quad \text{(2)} \]

### Step 3: Divide the equations
Dividing equation (1) by equation (2):
\[ \frac{2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)}{2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right)} = \frac{\frac{1}{4}}{\frac{1}{3}} \]
This simplifies to:
\[ \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{1/4}{1/3} = \frac{3}{4} \]

### Step 4: Find \( \sin(\alpha + \beta) \)
Let \( \theta = \frac{\alpha + \beta}{2} \). Then:
\[ \tan \theta = \frac{3}{4} \]
Using the identity for sine:
\[ \sin(2\theta) = \frac{2 \tan \theta}{1 + \tan^2 \theta} \]
Substituting \( \tan \theta = \frac{3}{4} \):
\[ \sin(2\theta) = \frac{2 \cdot \frac{3}{4}}{1 + \left(\frac{3}{4}\right)^2} = \frac{\frac{3}{2}}{1 + \frac{9}{16}} = \frac{\frac{3}{2}}{\frac{25}{16}} = \frac{3 \cdot 16}{2 \cdot 25} = \frac{48}{50} = \frac{24}{25} \]

### Step 5: Find \( \cos(\alpha + \beta) \)
Using the identity \( \cos^2(\alpha + \beta) = 1 - \sin^2(\alpha + \beta) \):
\[ \cos^2(\alpha + \beta) = 1 - \left(\frac{24}{25}\right)^2 = 1 - \frac{576}{625} = \frac{625 - 576}{625} = \frac{49}{625} \]
Taking the square root:
\[ \cos(\alpha + \beta) = \sqrt{\frac{49}{625}} = \frac{7}{25} \]

### Final Answer
Thus, the value of \( \cos(\alpha + \beta) \) is:
\[ \boxed{\frac{7}{25}} \]