If `tan2theta.tan4theta=1` , then the value of `tan3theta` is

Options

A. `sqrt(3)`
B. 0
C. 1
D. `(1)/(sqrt(3))`

Answer

To solve the problem where \( \tan 2\theta \cdot \tan 4\theta = 1 \) and we need to find the value of \( \tan 3\theta \), we can follow these steps:

### Step 1: Use the Identity
We know that if \( \tan A \cdot \tan B = 1 \), then \( A + B = 90^\circ \). Here, we can set \( A = 2\theta \) and \( B = 4\theta \).

### Step 2: Set Up the Equation
From the identity, we have:
\[ 2\theta + 4\theta = 90^\circ \]
This simplifies to:
\[ 6\theta = 90^\circ \]

### Step 3: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ \theta = \frac{90^\circ}{6} = 15^\circ \]

### Step 4: Find \( \tan 3\theta \)
Next, we need to find \( \tan 3\theta \):
\[ 3\theta = 3 \times 15^\circ = 45^\circ \]
Now, we calculate \( \tan 45^\circ \):
\[ \tan 45^\circ = 1 \]

### Conclusion
Thus, the value of \( \tan 3\theta \) is:
\[ \boxed{1} \]