If the sum of all value of x satisfying the system of equations
`tan x + tan y+ tan x* tan y=5 `
`sin (x +y)=4 cos x * cos y`
is `(k pi )/2` , where ` x in (0, (pi)/(2))` then find the values of k .

Answer

To solve the given system of equations:

1. **Equations**:
- \( \tan x + \tan y + \tan x \tan y = 5 \) (1)
- \( \sin(x+y) = 4 \cos x \cos y \) (2)

2. **Using the identity for sine**:
From equation (2), we know that:
\[ \sin(x+y) = \sin x \cos y + \cos x \sin y \]
Therefore, we can rewrite equation (2) as:
\[ \sin x \cos y + \cos x \sin y = 4 \cos x \cos y \]
Dividing both sides by \( \cos x \cos y \) (assuming \( \cos x, \cos y \neq 0 \)):
\[ \frac{\sin x}{\cos x} + \frac{\sin y}{\cos y} = 4 \]
This simplifies to:
\[ \tan x + \tan y = 4 \quad (3) \]

3. **Substituting equation (3) into equation (1)**:
Now substituting \( \tan x + \tan y = 4 \) into equation (1):
\[ 4 + \tan x \tan y = 5 \]
This gives us:
\[ \tan x \tan y = 1 \quad (4) \]

4. **Using the equations from (3) and (4)**:
We have two equations now:
- \( \tan x + \tan y = 4 \) (from (3))
- \( \tan x \tan y = 1 \) (from (4))

Let \( \tan x = a \) and \( \tan y = b \). Then we can write:
\[ a + b = 4 \]
\[ ab = 1 \]

5. **Forming a quadratic equation**:
The equations can be used to form a quadratic:
\[ t^2 - (a+b)t + ab = 0 \]
Substituting the values:
\[ t^2 - 4t + 1 = 0 \]

6. **Finding the roots**:
We can find the roots using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -4, c = 1 \):
\[ t = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]
Thus, the roots are \( \tan x_1 = 2 + \sqrt{3} \) and \( \tan x_2 = 2 - \sqrt{3} \).

7. **Finding the angles**:
To find \( x_1 \) and \( x_2 \):
\[ x_1 = \tan^{-1}(2 + \sqrt{3}), \quad x_2 = \tan^{-1}(2 - \sqrt{3}) \]

8. **Finding the sum of solutions**:
The sum of the angles \( x_1 + x_2 \) can be calculated using the identity:
\[ x_1 + x_2 = \tan^{-1}(2 + \sqrt{3}) + \tan^{-1}(2 - \sqrt{3}) = \frac{\pi}{2} \]

9. **Final result**:
Since the sum of all values of \( x \) satisfying the equations is \( \frac{\pi}{2} \), we can express this as:
\[ \frac{k \pi}{2} \quad \text{where } k = 1 \]

Thus, the value of \( k \) is \( \boxed{1} \).