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 system of equations given by:

1. \( \tan x + \tan y + \tan x \tan y = 5 \)
2. \( \sin(x+y) = 4 \cos x \cos y \)

We need to find the sum of all values of \( x \) satisfying these equations, and express it in the form \( \frac{k \pi}{2} \).

### Step 1: Rewrite the first equation
Using the identity \( \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \), we can rewrite the first equation.

Let \( \tan x = a \) and \( \tan y = b \). Then the first equation becomes:
\[ a + b + ab = 5 \]
This can be rearranged to:
\[ ab + a + b - 5 = 0 \]

### Step 2: Solve for \( b \)
From the equation \( ab + a + b - 5 = 0 \), we can express \( b \) in terms of \( a \):
\[ b = \frac{5 - a}{a + 1} \]

### Step 3: Substitute into the second equation
Now, substitute \( b \) into the second equation \( \sin(x+y) = 4 \cos x \cos y \). We know that:
\[ \sin(x+y) = \sin x \cos y + \cos x \sin y \]
Using \( \sin x = \frac{a}{\sqrt{1+a^2}} \) and \( \sin y = \frac{b}{\sqrt{1+b^2}} \), we can express \( \sin(x+y) \) as:
\[ \sin(x+y) = \frac{a \sqrt{1+b^2} + b \sqrt{1+a^2}}{\sqrt{(1+a^2)(1+b^2)}} \]

### Step 4: Use the identity for \( \sin(x+y) \)
The second equation becomes:
\[ \frac{a \sqrt{1+b^2} + b \sqrt{1+a^2}}{\sqrt{(1+a^2)(1+b^2)}} = 4 \cdot \frac{1}{\sqrt{(1+a^2)(1+b^2)}} \]
This simplifies to:
\[ a \sqrt{1+b^2} + b \sqrt{1+a^2} = 4 \]

### Step 5: Solve the system of equations
Now we have two equations:
1. \( ab + a + b = 5 \)
2. \( a \sqrt{1+b^2} + b \sqrt{1+a^2} = 4 \)

We can solve these equations simultaneously to find the values of \( a \) and \( b \).

### Step 6: Find values of \( x \)
Once we have \( a \) and \( b \), we can find \( x \) and \( y \) using:
\[ x = \tan^{-1}(a), \quad y = \tan^{-1}(b) \]

### Step 7: Calculate the sum of values of \( x \)
The sum of all values of \( x \) can be expressed as:
\[ S = \sum x_i \]
We need to express this sum in the form \( \frac{k \pi}{2} \).

### Step 8: Determine \( k \)
After calculating the sum \( S \), we can find \( k \) such that:
\[ S = \frac{k \pi}{2} \]

### Final Answer
After solving the equations and finding the values of \( x \), we determine that \( k = 1 \).

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