Let a,b, c are the sides opposite to angles A, B , C respectively in a `Delta ABC tan""(A-B)/(2)=(a-b)/(a+b)cot ""C/2 and (a)/(sin A)=(b)/(sin B) =(c)/(sin C),`
If `a=6,b=3 and cos (A-B) =4/5`
Angle C is equal to

Options

A. `pi/4`
B. `pi/2`
C. `(3pi)/(4)`
D. `(2pi)/(3)`

Answer

To solve the problem, we need to find the angle \( C \) in triangle \( ABC \) given the sides \( a = 6 \), \( b = 3 \), and \( \cos(A - B) = \frac{4}{5} \). We will use the relation provided in the question:

\[ \tan\left(\frac{A - B}{2}\right) = \frac{a - b}{a + b} \cot\left(\frac{C}{2}\right) \]

### Step 1: Calculate \( \tan\left(\frac{A - B}{2}\right) \)

Using the identity for \( \tan\left(\frac{A - B}{2}\right) \):

\[ \tan\left(\frac{A - B}{2}\right) = \frac{\sin(A - B)}{1 + \cos(A - B)} \]

We know that \( \cos(A - B) = \frac{4}{5} \). To find \( \sin(A - B) \), we can use the Pythagorean identity:

\[ \sin^2(A - B) + \cos^2(A - B) = 1 \]

Substituting \( \cos(A - B) \):

\[ \sin^2(A - B) + \left(\frac{4}{5}\right)^2 = 1 \]

Calculating:

\[ \sin^2(A - B) + \frac{16}{25} = 1 \]

\[ \sin^2(A - B) = 1 - \frac{16}{25} = \frac{9}{25} \]

Thus, \( \sin(A - B) = \frac{3}{5} \).

Now substituting back into the equation for \( \tan\left(\frac{A - B}{2}\right) \):

\[ \tan\left(\frac{A - B}{2}\right) = \frac{\frac{3}{5}}{1 + \frac{4}{5}} = \frac{\frac{3}{5}}{\frac{9}{5}} = \frac{3}{9} = \frac{1}{3} \]

### Step 2: Set up the equation with the given relation

Now we can substitute \( \tan\left(\frac{A - B}{2}\right) \) into the relation:

\[ \frac{1}{3} = \frac{a - b}{a + b} \cot\left(\frac{C}{2}\right) \]

Substituting \( a = 6 \) and \( b = 3 \):

\[ \frac{1}{3} = \frac{6 - 3}{6 + 3} \cot\left(\frac{C}{2}\right) \]

Calculating:

\[ \frac{1}{3} = \frac{3}{9} \cot\left(\frac{C}{2}\right) = \frac{1}{3} \cot\left(\frac{C}{2}\right) \]

### Step 3: Solve for \( \cot\left(\frac{C}{2}\right) \)

Cross-multiplying gives:

\[ 1 = \cot\left(\frac{C}{2}\right) \]

This implies:

\[ \cot\left(\frac{C}{2}\right) = 1 \]

### Step 4: Find \( C \)

Since \( \cot\left(\frac{C}{2}\right) = 1 \), we have:

\[ \frac{C}{2} = \frac{\pi}{4} \]

Thus, multiplying both sides by 2 gives:

\[ C = \frac{\pi}{2} \]

### Conclusion

The angle \( C \) is equal to \( \frac{\pi}{2} \).