If `tan((theta)/(2))=5/2and tan((phi)/(2))=3/4,` the value of `cos(theta+phi),` is

Options

A. `-(364)/(725)`
B. `-(627)/(725)`
C. `-(240)/(339)`
D. `-(339)/(725)`

Answer

To solve the problem, we need to find the value of \( \cos(\theta + \phi) \) given that \( \tan\left(\frac{\theta}{2}\right) = \frac{5}{2} \) and \( \tan\left(\frac{\phi}{2}\right) = \frac{3}{4} \).

### Step-by-Step Solution:

1. **Use the tangent addition formula**:
The formula for \( \tan(A + B) \) is:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Here, let \( A = \frac{\theta}{2} \) and \( B = \frac{\phi}{2} \). Thus,
\[ \tan\left(\frac{\theta + \phi}{2}\right) = \frac{\tan\left(\frac{\theta}{2}\right) + \tan\left(\frac{\phi}{2}\right)}{1 - \tan\left(\frac{\theta}{2}\right) \tan\left(\frac{\phi}{2}\right)} \]

2. **Substitute the values**:
Substitute \( \tan\left(\frac{\theta}{2}\right) = \frac{5}{2} \) and \( \tan\left(\frac{\phi}{2}\right) = \frac{3}{4} \):
\[ \tan\left(\frac{\theta + \phi}{2}\right) = \frac{\frac{5}{2} + \frac{3}{4}}{1 - \left(\frac{5}{2} \cdot \frac{3}{4}\right)} \]

3. **Calculate the numerator**:
To add \( \frac{5}{2} \) and \( \frac{3}{4} \), find a common denominator:
\[ \frac{5}{2} = \frac{10}{4} \quad \text{(convert to a common denominator)} \]
Now, add:
\[ \frac{10}{4} + \frac{3}{4} = \frac{13}{4} \]

4. **Calculate the denominator**:
Now calculate \( 1 - \left(\frac{5}{2} \cdot \frac{3}{4}\right) \):
\[ \frac{5}{2} \cdot \frac{3}{4} = \frac{15}{8} \]
Therefore,
\[ 1 - \frac{15}{8} = \frac{8}{8} - \frac{15}{8} = -\frac{7}{8} \]

5. **Combine the results**:
Now substitute back into the tangent formula:
\[ \tan\left(\frac{\theta + \phi}{2}\right) = \frac{\frac{13}{4}}{-\frac{7}{8}} = \frac{13}{4} \cdot -\frac{8}{7} = -\frac{26}{7} \]

6. **Find \( \cos(\theta + \phi) \)**:
Use the identity:
\[ \cos(x) = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]
Here, \( x = \theta + \phi \):
\[ \cos(\theta + \phi) = \frac{1 - \tan^2\left(\frac{\theta + \phi}{2}\right)}{1 + \tan^2\left(\frac{\theta + \phi}{2}\right)} \]
Substitute \( \tan\left(\frac{\theta + \phi}{2}\right) = -\frac{26}{7} \):
\[ \tan^2\left(\frac{\theta + \phi}{2}\right) = \left(-\frac{26}{7}\right)^2 = \frac{676}{49} \]

7. **Calculate \( \cos(\theta + \phi) \)**:
Substitute into the cosine formula:
\[ \cos(\theta + \phi) = \frac{1 - \frac{676}{49}}{1 + \frac{676}{49}} = \frac{\frac{49}{49} - \frac{676}{49}}{\frac{49}{49} + \frac{676}{49}} = \frac{\frac{49 - 676}{49}}{\frac{49 + 676}{49}} = \frac{-627}{725} \]

### Final Answer:
\[ \cos(\theta + \phi) = -\frac{627}{725} \]