If `(1 - tan x)/(1 + tan x) = tan y and x - y = pi/6` , then x, y are respectively

Options

A. `(5pi)/24, pi/24`
B. `-(7pi)/24, -(11pi)/24`
C. `-(115pi)/24- (119pi)/24`
D. none

Answer

To solve the equation \((1 - \tan x)/(1 + \tan x) = \tan y\) given that \(x - y = \frac{\pi}{6}\), we can follow these steps:

### Step 1: Use the tangent subtraction formula
We know that:
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
We can rewrite the left side of the equation using the tangent subtraction formula:
\[ \frac{1 - \tan x}{1 + \tan x} = \tan\left(\frac{\pi}{4} - x\right) \]
Thus, we can equate:
\[ \tan\left(\frac{\pi}{4} - x\right) = \tan y \]

### Step 2: Set the angles equal
Since the tangent function is periodic, we can set the angles equal to each other:
\[ \frac{\pi}{4} - x = y + n\pi \quad (n \in \mathbb{Z}) \]
For simplicity, we will consider \(n = 0\):
\[ \frac{\pi}{4} - x = y \]

### Step 3: Substitute \(y\) in the given equation
We have two equations now:
1. \(x - y = \frac{\pi}{6}\)
2. \(\frac{\pi}{4} - x = y\)

Substituting \(y\) from the second equation into the first:
\[ x - \left(\frac{\pi}{4} - x\right) = \frac{\pi}{6} \]
This simplifies to:
\[ x + x - \frac{\pi}{4} = \frac{\pi}{6} \]
\[ 2x - \frac{\pi}{4} = \frac{\pi}{6} \]

### Step 4: Solve for \(x\)
To solve for \(x\), we first find a common denominator for \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\):
The least common multiple of 4 and 6 is 12. Thus:
\[ \frac{\pi}{4} = \frac{3\pi}{12}, \quad \frac{\pi}{6} = \frac{2\pi}{12} \]
Substituting these values gives:
\[ 2x - \frac{3\pi}{12} = \frac{2\pi}{12} \]
Adding \(\frac{3\pi}{12}\) to both sides:
\[ 2x = \frac{2\pi}{12} + \frac{3\pi}{12} = \frac{5\pi}{12} \]
Dividing by 2:
\[ x = \frac{5\pi}{24} \]

### Step 5: Find \(y\)
Now substituting \(x\) back into the equation for \(y\):
\[ y = \frac{\pi}{4} - x = \frac{\pi}{4} - \frac{5\pi}{24} \]
Finding a common denominator for \(\frac{\pi}{4}\):
\[ \frac{\pi}{4} = \frac{6\pi}{24} \]
Thus:
\[ y = \frac{6\pi}{24} - \frac{5\pi}{24} = \frac{\pi}{24} \]

### Final Result
The values of \(x\) and \(y\) are:
\[ x = \frac{5\pi}{24}, \quad y = \frac{\pi}{24} \]