Let a = 6, b = 3 and `cos (A -B) = (4)/(5)`
Angle C is equal to

Options

A. `(3pi)/(4)`
B. `(pi)/(4)`
C. `(pi)/(2)`
D. none of these

Explanation

To solve the problem step-by-step, we will use the given values of \( a \), \( b \), and \( \cos(A - B) \) to find angle \( C \).

### Step 1: Write down the given information
We are given:
- \( a = 6 \)
- \( b = 3 \)
- \( \cos(A - B) = \frac{4}{5} \)

### Step 2: Use the formula for \( \cos(A - B) \)
We know that:
\[ \cos(A - B) = \frac{1 - \tan^2\left(\frac{A - B}{2}\right)}{1 + \tan^2\left(\frac{A - B}{2}\right)} \]
Setting this equal to \( \frac{4}{5} \):
\[ \frac{1 - \tan^2\left(\frac{A - B}{2}\right)}{1 + \tan^2\left(\frac{A - B}{2}\right)} = \frac{4}{5} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5(1 - \tan^2\left(\frac{A - B}{2}\right)) = 4(1 + \tan^2\left(\frac{A - B}{2}\right)) \]

### Step 4: Expand and rearrange the equation
Expanding both sides:
\[ 5 - 5\tan^2\left(\frac{A - B}{2}\right) = 4 + 4\tan^2\left(\frac{A - B}{2}\right) \]
Rearranging gives:
\[ 5 - 4 = 5\tan^2\left(\frac{A - B}{2}\right) + 4\tan^2\left(\frac{A - B}{2}\right) \]
\[ 1 = 9\tan^2\left(\frac{A - B}{2}\right) \]

### Step 5: Solve for \( \tan\left(\frac{A - B}{2}\right) \)
Dividing both sides by 9:
\[ \tan^2\left(\frac{A - B}{2}\right) = \frac{1}{9} \]
Taking the square root:
\[ \tan\left(\frac{A - B}{2}\right) = \frac{1}{3} \]

### Step 6: Use the formula for \( \tan\left(\frac{A - B}{2}\right) \)
We know:
\[ \tan\left(\frac{A - B}{2}\right) = \frac{A - B}{A + B} \cdot \cot\left(\frac{C}{2}\right) \]
Substituting the known values:
\[ \frac{1}{3} = \frac{6 - 3}{6 + 3} \cdot \cot\left(\frac{C}{2}\right) \]
This simplifies to:
\[ \frac{1}{3} = \frac{3}{9} \cdot \cot\left(\frac{C}{2}\right) \]
\[ \frac{1}{3} = \frac{1}{3} \cdot \cot\left(\frac{C}{2}\right) \]

### Step 7: Solve for \( \cot\left(\frac{C}{2}\right) \)
From the equation:
\[ 1 = \cot\left(\frac{C}{2}\right) \]
This means:
\[ \cot\left(\frac{C}{2}\right) = 1 \]

### Step 8: Find angle \( C \)
Since \( \cot\left(\frac{C}{2}\right) = 1 \), we have:
\[ \frac{C}{2} = 45^\circ \quad \text{or} \quad \frac{C}{2} = \frac{\pi}{4} \]
Thus:
\[ C = 90^\circ \quad \text{or} \quad C = \frac{\pi}{2} \]

### Final Answer
Therefore, angle \( C \) is:
\[ C = \frac{\pi}{2} \]