In a `DeltaABC, if tanA+2tanB=0,` then which one is correct?

Options

A. `0lttan^(2)Cle1/8`
B. `1/8lttan^(2)Cle1/2`
C. `1/2lttan^(2)Clt1`
D. none of these

Explanation

To solve the problem, we need to analyze the given equation in the context of triangle ABC, where \( \tan A + 2 \tan B = 0 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \tan A + 2 \tan B = 0 \]
From this, we can express \( \tan A \) in terms of \( \tan B \):
\[ \tan A = -2 \tan B \]

**Hint:** Rearranging the equation helps isolate one variable.

2. **Use the identity for the sum of tangents in a triangle:**
In any triangle, the following identity holds:
\[ \tan A + \tan B + \tan C = \tan A \tan B \tan C \]
Substitute \( \tan A = -2 \tan B \) into this identity:
\[ -2 \tan B + \tan B + \tan C = (-2 \tan B) \tan B \tan C \]

**Hint:** Remember that the sum of angles in a triangle is related to their tangents.

3. **Simplify the equation:**
This simplifies to:
\[ -\tan B + \tan C = -2 \tan^2 B \tan C \]
Rearranging gives:
\[ \tan C + 2 \tan^2 B \tan C = \tan B \]
Factor out \( \tan C \):
\[ \tan C (1 + 2 \tan^2 B) = \tan B \]

**Hint:** Factoring can help simplify complex equations.

4. **Rearranging gives:**
\[ \tan C = \frac{\tan B}{1 + 2 \tan^2 B} \]

**Hint:** Isolating \( \tan C \) allows us to analyze its behavior as \( \tan B \) changes.

5. **Finding the bounds for \( \tan^2 C \):**
Since \( \tan C = \frac{\tan B}{1 + 2 \tan^2 B} \), we can square both sides:
\[ \tan^2 C = \left( \frac{\tan B}{1 + 2 \tan^2 B} \right)^2 \]

**Hint:** Squaring the equation helps in finding the range of \( \tan^2 C \).

6. **Analyze the limits:**
As \( \tan B \) approaches 0, \( \tan^2 C \) approaches 0. As \( \tan B \) increases, we need to analyze the behavior of \( \tan^2 C \):
\[ \tan^2 C \text{ will be maximized when } 1 + 2 \tan^2 B \text{ is minimized.} \]
The minimum occurs when \( \tan^2 B = 0 \), leading to:
\[ \tan^2 C \leq \frac{1}{8} \]

**Hint:** Understanding limits and behavior of functions helps in determining maximum and minimum values.

7. **Conclusion:**
Thus, we find that:
\[ 0 \leq \tan^2 C \leq \frac{1}{8} \]
Therefore, the correct option is:
\[ \text{The value of } \tan^2 C \text{ goes from } 0 \text{ to } \frac{1}{8}. \]

**Final Answer:** The correct option is that \( \tan^2 C \) values go from \( 0 \) to \( \frac{1}{8} \).