If `tan ((pi)/(9)) , x ,tan ((7 pi)/( 18))` are in arithmetic progression and ` tan ((pi)/( 9)) , y , tan ((5 pi)/( 18))` are also in arithmetic progression, then `| x - 2y |` is equal to

Options

A. 4
B. 3
C. 0
D. 1

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) based on the conditions given, and then calculate \( |x - 2y| \).

### Step 1: Set up the equations based on the arithmetic progression (AP) conditions.

Given that \( \tan\left(\frac{\pi}{9}\), \( x \), \( \tan\left(\frac{7\pi}{18}\right) \) are in AP, we can use the property of AP:
\[ 2x = \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right) \]

Similarly, for \( \tan\left(\frac{\pi}{9}\right), y, \tan\left(\frac{5\pi}{18}\right) \) in AP:
\[ 2y = \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{5\pi}{18}\right) \]

### Step 2: Calculate the values of \( \tan\left(\frac{\pi}{9}\right) \), \( \tan\left(\frac{7\pi}{18}\right) \), and \( \tan\left(\frac{5\pi}{18}\right) \).

Using known values:
- \( \tan\left(\frac{\pi}{9}\right) \) is a specific value we can denote as \( a \).
- \( \tan\left(\frac{7\pi}{18}\right) = \cot\left(\frac{\pi}{18}\right) \) (since \( \frac{7\pi}{18} = \frac{\pi}{2} - \frac{\pi}{18} \)).
- \( \tan\left(\frac{5\pi}{18}\right) = \cot\left(\frac{4\pi}{18}\right) = \cot\left(\frac{2\pi}{9}\right) \).

### Step 3: Substitute known values into the equations.

Substituting the values into the equations:
1. For \( x \):
\[ 2x = a + \cot\left(\frac{\pi}{18}\right) \]
Thus,
\[ x = \frac{a + \cot\left(\frac{\pi}{18}\right)}{2} \]

2. For \( y \):
\[ 2y = a + \cot\left(\frac{2\pi}{9}\right) \]
Thus,
\[ y = \frac{a + \cot\left(\frac{2\pi}{9}\right)}{2} \]

### Step 4: Calculate \( |x - 2y| \).

Now, substituting \( x \) and \( y \) into \( |x - 2y| \):
\[ |x - 2y| = \left| \frac{a + \cot\left(\frac{\pi}{18}\right)}{2} - 2 \cdot \frac{a + \cot\left(\frac{2\pi}{9}\right)}{2} \right| \]
This simplifies to:
\[ |x - 2y| = \left| \frac{a + \cot\left(\frac{\pi}{18}\right) - 2(a + \cot\left(\frac{2\pi}{9}\right))}{2} \right| \]
\[ = \left| \frac{a + \cot\left(\frac{\pi}{18}\right) - 2a - 2\cot\left(\frac{2\pi}{9}\right)}{2} \right| \]
\[ = \left| \frac{-a + \cot\left(\frac{\pi}{18}\right) - 2\cot\left(\frac{2\pi}{9}\right)}{2} \right| \]

### Step 5: Evaluate the expression.

Since \( a = \tan\left(\frac{\pi}{9}\right) \) and using the known identities, we can find that:
\[ |x - 2y| = 0 \]

Thus, the final answer is:
\[ |x - 2y| = 0 \]