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

Options

A. `(25)/(7)`
B. `(25)/(12)`
C. `(25)/(13)`
D. `(24)/(7)`

Explanation

To find the value of \( \tan(\alpha + \beta) \) 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 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 two 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:
\[ \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{3}{4} \]

### Step 4: Find \( \tan(\alpha + \beta) \)
Let \( \theta = \frac{\alpha + \beta}{2} \). Then \( \tan(\theta) = \frac{3}{4} \).
Using the double angle formula for tangent:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \( \tan(\theta) = \frac{3}{4} \):
\[ \tan(2\theta) = \frac{2 \cdot \frac{3}{4}}{1 - \left(\frac{3}{4}\right)^2} \]
Calculating \( \tan(2\theta) \):
\[ = \frac{\frac{6}{4}}{1 - \frac{9}{16}} = \frac{\frac{6}{4}}{\frac{7}{16}} = \frac{6}{4} \cdot \frac{16}{7} = \frac{24}{7} \]

### Final Answer
Thus, the value of \( \tan(\alpha + \beta) \) is:
\[ \boxed{\frac{24}{7}} \]