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

Options

A. `x gt y`
B. `x =y`
C. `2x =y`
D. `x =2y`

Answer

To solve the problem, we need to find the arithmetic means (A.M.) of the given tangent values and then establish the relationship between them.

### Step-by-step Solution:

1. **Identify the Values:**
We have:
- \( a = \tan\left(\frac{\pi}{9}\right) \)
- \( b = \tan\left(\frac{5\pi}{18}\right) \)
- \( c = \tan\left(\frac{7\pi}{18}\right) \)

2. **Calculate x:**
The arithmetic mean \( x \) of \( a \) and \( b \) is given by:
\[ x = \frac{a + b}{2} = \frac{\tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{5\pi}{18}\right)}{2} \]

3. **Calculate y:**
The arithmetic mean \( y \) of \( a \) and \( c \) is given by:
\[ y = \frac{a + c}{2} = \frac{\tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right)}{2} \]

4. **Convert Angles to Degrees:**
- \( \frac{\pi}{9} = 20^\circ \)
- \( \frac{5\pi}{18} = 50^\circ \)
- \( \frac{7\pi}{18} = 70^\circ \)

5. **Substitute Values:**
Now substituting the angles into the expressions for \( x \) and \( y \):
\[ x = \frac{\tan(20^\circ) + \tan(50^\circ)}{2} \]
\[ y = \frac{\tan(20^\circ) + \tan(70^\circ)}{2} \]

6. **Use the Tangent Addition Formula:**
We know from trigonometric identities that:
\[ \tan(70^\circ) = \cot(20^\circ) = \frac{1}{\tan(20^\circ)} \]
Therefore, we can express \( y \) as:
\[ y = \frac{\tan(20^\circ) + \frac{1}{\tan(20^\circ)}}{2} \]

7. **Simplify y:**
Let \( t = \tan(20^\circ) \):
\[ y = \frac{t + \frac{1}{t}}{2} = \frac{t^2 + 1}{2t} \]

8. **Compare x and y:**
Now we compare \( x \) and \( y \):
- \( x = \frac{\tan(20^\circ) + \tan(50^\circ)}{2} \)
- \( y = \frac{\tan(20^\circ) + \frac{1}{\tan(20^\circ)}}{2} \)

Since \( \tan(50^\circ) > \tan(20^\circ) \), it follows that \( x < y \).

9. **Conclusion:**
Thus, we can conclude that:
\[ y > x \]
This implies that \( 2x < y \) or \( y = 2x \) is a possible relationship.