If `cos theta = 3/5 and cos phi =4/5,` where `theta` and `phi` are positive acute angles, then `cos ""(theta - phi)/(2)=`

Options

A. `(7)/(sqrt2)`
B. `(7)/(5sqrt2)`
C. `(7)/(sqrt5)`
D. `(7)/(2sqrt5)`

Answer

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

### Step-by-Step Solution:

1. **Find \( \sin \theta \) and \( \sin \phi \)**:
- We know that \( \sin^2 \theta + \cos^2 \theta = 1 \).
- Therefore, \( \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \).
- Thus, \( \sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \).

- Similarly for \( \phi \):
- \( \sin^2 \phi = 1 - \cos^2 \phi = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \).
- Thus, \( \sin \phi = \sqrt{\frac{9}{25}} = \frac{3}{5} \).

2. **Use the cosine of the difference formula**:
- The formula for \( \cos(\theta - \phi) \) is:
\[ \cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi \]
- Substitute the values:
\[ \cos(\theta - \phi) = \left(\frac{3}{5}\right)\left(\frac{4}{5}\right) + \left(\frac{4}{5}\right)\left(\frac{3}{5}\right) \]
- This simplifies to:
\[ \cos(\theta - \phi) = \frac{12}{25} + \frac{12}{25} = \frac{24}{25} \]

3. **Use the half-angle formula**:
- The half-angle formula for cosine is:
\[ \cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + \cos x}{2}} \]
- Here, let \( x = \theta - \phi \):
\[ \cos\left(\frac{\theta - \phi}{2}\right) = \sqrt{\frac{1 + \cos(\theta - \phi)}{2}} = \sqrt{\frac{1 + \frac{24}{25}}{2}} \]
- Simplifying this:
\[ \cos\left(\frac{\theta - \phi}{2}\right) = \sqrt{\frac{\frac{49}{25}}{2}} = \sqrt{\frac{49}{50}} = \frac{7}{\sqrt{50}} = \frac{7\sqrt{2}}{10} \]

### Final Answer:
\[ \cos\left(\frac{\theta - \phi}{2}\right) = \frac{7\sqrt{2}}{10} \]