If X is `A.M.` of `tan(pi/9)` and `tan ((5pi)/18)` and `y` is `A.M.` of `tan(pi/9)` and `tan((7pi)/18)`, then

Options

A. x > y
B. x = y
C. y = 2x
D. x = 2y

Explanation

To solve the problem, we need to calculate the arithmetic means \( x \) and \( y \) and then compare them.

### Step 1: Calculate \( x \)
The arithmetic mean \( x \) of \( \tan\left(\frac{\pi}{9}\right) \) and \( \tan\left(\frac{5\pi}{18}\right) \) is given by:

\[ x = \frac{\tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{5\pi}{18}\right)}{2} \]

### Step 2: Simplify \( \tan\left(\frac{5\pi}{18}\right) \)
We can express \( \tan\left(\frac{5\pi}{18}\right) \) using the identity for tangent of a sum:

\[ \tan\left(\frac{5\pi}{18}\right) = \tan\left(\frac{\pi}{2} - \frac{\pi}{9}\right) = \cot\left(\frac{\pi}{9}\right) = \frac{1}{\tan\left(\frac{\pi}{9}\right)} \]

### Step 3: Substitute into the expression for \( x \)
Now substituting this back into the equation for \( x \):

\[ x = \frac{\tan\left(\frac{\pi}{9}\right) + \frac{1}{\tan\left(\frac{\pi}{9}\right)}}{2} \]

### Step 4: Combine the terms
Let \( t = \tan\left(\frac{\pi}{9}\right) \). Then:

\[ x = \frac{t + \frac{1}{t}}{2} = \frac{\frac{t^2 + 1}{t}}{2} = \frac{t^2 + 1}{2t} \]

### Step 5: Calculate \( y \)
Now, we calculate the arithmetic mean \( y \) of \( \tan\left(\frac{\pi}{9}\right) \) and \( \tan\left(\frac{7\pi}{18}\right) \):

\[ y = \frac{\tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right)}{2} \]

### Step 6: Simplify \( \tan\left(\frac{7\pi}{18}\right) \)
Using the identity for tangent of a sum again:

\[ \tan\left(\frac{7\pi}{18}\right) = \tan\left(\frac{\pi}{2} - \frac{\pi}{9}\right) = \cot\left(\frac{\pi}{9}\right) = \frac{1}{\tan\left(\frac{\pi}{9}\right)} = \frac{1}{t} \]

### Step 7: Substitute into the expression for \( y \)
Now substituting this back into the equation for \( y \):

\[ y = \frac{\tan\left(\frac{\pi}{9}\right) + \frac{1}{\tan\left(\frac{\pi}{9}\right)}}{2} = \frac{t + \frac{1}{t}}{2} = \frac{t^2 + 1}{2t} \]

### Step 8: Compare \( x \) and \( y \)
From the calculations, we see that:

\[ x = y \]

### Conclusion
Thus, the final result is:

\[ y = 2x \quad \text{(since both x and y are equal)} \]

### Final Answer
The correct option is \( y = 2x \).