If `cos alpha = (3)/(5) , cos beta = (4)/(5) , ` find the value of ` cos ""(( alpha - beta)/( 2)),` assuming ` alpha and beta` to be acute angles.

Answer

To find the value of \( \cos\left(\frac{\alpha - \beta}{2}\right) \) given \( \cos \alpha = \frac{3}{5} \) and \( \cos \beta = \frac{4}{5} \), we can follow these steps:

### Step 1: Find \( \sin \alpha \) and \( \sin \beta \)

Since \( \alpha \) and \( \beta \) are acute angles, we can use the Pythagorean identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

For \( \alpha \):

\[ \sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]

Thus,

\[ \sin \alpha = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

For \( \beta \):

\[ \sin^2 \beta = 1 - \cos^2 \beta = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]

Thus,

\[ \sin \beta = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 2: Use the cosine difference formula

We can find \( \cos(\alpha - \beta) \) using the formula:

\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]

Substituting the values we found:

\[ \cos(\alpha - \beta) = \left(\frac{3}{5}\right) \left(\frac{4}{5}\right) + \left(\frac{4}{5}\right) \left(\frac{3}{5}\right) \]

Calculating this gives:

\[ \cos(\alpha - \beta) = \frac{12}{25} + \frac{12}{25} = \frac{24}{25} \]

### Step 3: Use the half-angle formula

Now, we can find \( \cos\left(\frac{\alpha - \beta}{2}\right) \) using the half-angle formula:

\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} \]

Here, \( \theta = \alpha - \beta \), so:

\[ \cos\left(\frac{\alpha - \beta}{2}\right) = \sqrt{\frac{1 + \cos(\alpha - \beta)}{2}} = \sqrt{\frac{1 + \frac{24}{25}}{2}} \]

Calculating this gives:

\[ \cos\left(\frac{\alpha - \beta}{2}\right) = \sqrt{\frac{\frac{49}{25}}{2}} = \sqrt{\frac{49}{50}} = \frac{7}{\sqrt{50}} = \frac{7\sqrt{2}}{10} \]

### Final Answer

Thus, the value of \( \cos\left(\frac{\alpha - \beta}{2}\right) \) is:

\[ \frac{7\sqrt{2}}{10} \]

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