If `tan(pi/9) , x , tan((7pi)/18)` are in A.P. and `tan(pi/9) , y , tan((5pi)/18)` are also in A.P then find `abs(x-2y)`

Options

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

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that \( \tan(\frac{\pi}{9}), x, \tan(\frac{7\pi}{18}) \) are in arithmetic progression (A.P.) and \( \tan(\frac{\pi}{9}), y, \tan(\frac{5\pi}{18}) \) are also in A.P. Finally, we will compute \( |x - 2y| \).

### Step-by-step Solution:

1. **Understanding A.P. Condition**:
If three terms \( a, b, c \) are in A.P., then the condition can be written as:
\[ 2b = a + c \]

2. **Applying A.P. Condition for \( x \)**:
For the first A.P. \( \tan(\frac{\pi}{9}), x, \tan(\frac{7\pi}{18}) \):
\[ 2x = \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right) \]
Therefore,
\[ x = \frac{1}{2} \left( \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right) \right) \]

3. **Applying A.P. Condition for \( y \)**:
For the second A.P. \( \tan(\frac{\pi}{9}), y, \tan(\frac{5\pi}{18}) \):
\[ 2y = \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{5\pi}{18}\right) \]
Therefore,
\[ y = \frac{1}{2} \left( \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{5\pi}{18}\right) \right) \]

4. **Finding \( x - 2y \)**:
Now we need to compute \( |x - 2y| \):
\[ x - 2y = \frac{1}{2} \left( \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right) \right) - 2 \cdot \frac{1}{2} \left( \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{5\pi}{18}\right) \right) \]
Simplifying this gives:
\[ x - 2y = \frac{1}{2} \left( \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right) - 2 \tan\left(\frac{\pi}{9}\right) - 2 \tan\left(\frac{5\pi}{18}\right) \right) \]
\[ = \frac{1}{2} \left( -\tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right) - 2 \tan\left(\frac{5\pi}{18}\right) \right) \]

5. **Using Tangent Addition Formula**:
We know that:
\[ \tan\left(\frac{7\pi}{18}\right) = \tan\left(\frac{\pi}{2} - \frac{\pi}{18}\right) = \cot\left(\frac{\pi}{18}\right) \]
Thus,
\[ \tan\left(\frac{7\pi}{18}\right) = \frac{1}{\tan\left(\frac{\pi}{18}\right)} \]

6. **Substituting Values**:
Substitute the values back into the equation and simplify:
\[ |x - 2y| = \left| \frac{1}{2} \left( -\tan\left(\frac{\pi}{9}\right) + \frac{1}{\tan\left(\frac{\pi}{18}\right)} - 2 \tan\left(\frac{5\pi}{18}\right) \right) \right| \]

7. **Final Calculation**:
After simplifying further, we find that the expression simplifies to zero:
\[ |x - 2y| = 0 \]

### Conclusion:
Thus, the final answer is:
\[ \boxed{0} \]