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 find the value of \( \cos(\theta + \phi) \) given \( \tan\left(\frac{\theta}{2}\right) = \frac{5}{2} \) and \( \tan\left(\frac{\phi}{2}\right) = \frac{3}{4} \), we will follow these steps:

### Step 1: Find \( \tan(\theta) \)
Using the double angle formula for tangent:
\[ \tan(\theta) = \frac{2 \tan\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)} \]
Substituting \( \tan\left(\frac{\theta}{2}\right) = \frac{5}{2} \):
\[ \tan(\theta) = \frac{2 \cdot \frac{5}{2}}{1 - \left(\frac{5}{2}\right)^2} = \frac{5}{1 - \frac{25}{4}} = \frac{5}{\frac{-21}{4}} = \frac{20}{-21} \]
Thus, \( \tan(\theta) = -\frac{20}{21} \).

### Step 2: Find \( \tan(\phi) \)
Using the same double angle formula:
\[ \tan(\phi) = \frac{2 \tan\left(\frac{\phi}{2}\right)}{1 - \tan^2\left(\frac{\phi}{2}\right)} \]
Substituting \( \tan\left(\frac{\phi}{2}\right) = \frac{3}{4} \):
\[ \tan(\phi) = \frac{2 \cdot \frac{3}{4}}{1 - \left(\frac{3}{4}\right)^2} = \frac{\frac{3}{2}}{1 - \frac{9}{16}} = \frac{\frac{3}{2}}{\frac{7}{16}} = \frac{3 \cdot 16}{2 \cdot 7} = \frac{24}{7} \]
Thus, \( \tan(\phi) = \frac{24}{7} \).

### Step 3: Find \( \cos(\theta) \) and \( \sin(\theta) \)
From \( \tan(\theta) = -\frac{20}{21} \):
- Let the opposite side be \( -20 \) and the adjacent side be \( 21 \).
- The hypotenuse \( h \) is given by:
\[ h = \sqrt{(-20)^2 + 21^2} = \sqrt{400 + 441} = \sqrt{841} = 29 \]
Thus,
\[ \sin(\theta) = \frac{-20}{29}, \quad \cos(\theta) = \frac{21}{29} \]

### Step 4: Find \( \cos(\phi) \) and \( \sin(\phi) \)
From \( \tan(\phi) = \frac{24}{7} \):
- Let the opposite side be \( 24 \) and the adjacent side be \( 7 \).
- The hypotenuse \( h \) is given by:
\[ h = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]
Thus,
\[ \sin(\phi) = \frac{24}{25}, \quad \cos(\phi) = \frac{7}{25} \]

### Step 5: Use the cosine addition formula
Now we can find \( \cos(\theta + \phi) \):
\[ \cos(\theta + \phi) = \cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi) \]
Substituting the values we found:
\[ \cos(\theta + \phi) = \left(\frac{21}{29}\right)\left(\frac{7}{25}\right) - \left(-\frac{20}{29}\right)\left(\frac{24}{25}\right) \]
Calculating each term:
\[ \cos(\theta + \phi) = \frac{147}{725} + \frac{480}{725} = \frac{627}{725} \]
Thus, the final answer is:
\[ \cos(\theta + \phi) = -\frac{627}{725} \]

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