If `x+y+z=pi,,tan x tanz=2and tanytanz=18,then tan^(2)z=`

Options

A. 15
B. 16
C. 19
D. 20

Explanation

To solve the problem step by step, we start with the given equations:

1. **Given Equations:**
\[ x + y + z = \pi \]
\[ \tan x \tan z = 2 \]
\[ \tan y \tan z = 18 \]

2. **Rearranging the First Equation:**
From the first equation, we can express \(y\) in terms of \(x\) and \(z\):
\[ y = \pi - x - z \]

3. **Using the Tangent Addition Formula:**
We know that:
\[ \tan(\pi - \theta) = -\tan(\theta) \]
Therefore, we can write:
\[ \tan y = -\tan(x + z) = -\frac{\tan x + \tan z}{1 - \tan x \tan z} \]

4. **Substituting Values:**
Now, substituting \(\tan x \tan z = 2\) into the equation:
\[ \tan y = -\frac{\tan x + \tan z}{1 - 2} \]
This simplifies to:
\[ \tan y = \frac{\tan x + \tan z}{1} \]

5. **Using the Second Given Equation:**
From the second equation, we know:
\[ \tan y \tan z = 18 \]
Substituting \(\tan y\) from the previous step:
\[ \left(\tan x + \tan z\right) \tan z = 18 \]

6. **Letting \(\tan z = t\):**
Let \(\tan z = t\) and \(\tan x = a\). Then we can rewrite the equations:
\[ (a + t)t = 18 \quad \text{(1)} \]
\[ at = 2 \quad \text{(2)} \]

7. **From Equation (2):**
We can express \(a\) in terms of \(t\):
\[ a = \frac{2}{t} \]

8. **Substituting \(a\) into Equation (1):**
Substitute \(a\) back into Equation (1):
\[ \left(\frac{2}{t} + t\right)t = 18 \]
Simplifying this gives:
\[ 2 + t^2 = 18 \]
\[ t^2 = 16 \]

9. **Finding \(\tan^2 z\):**
Therefore, we find:
\[ \tan^2 z = t^2 = 16 \]

10. **Final Answer:**
Thus, the value of \(\tan^2 z\) is:
\[ \tan^2 z = 16 \]