If `tan 2 theta.tan 4 theta =1`, then the value of `3 theta` is

Options

A. `sqrt3`
B. 0
C. 1
D. `1/sqrt3`

Answer

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

### Step 1: Use the identity for tangent
We know that if \( \tan A \cdot \tan B = 1 \), then \( A + B = 90^\circ \) (or \( A + B = \frac{\pi}{2} \) in radians).

In our case, let:
- \( A = 2\theta \)
- \( B = 4\theta \)

Thus, we can write:
\[ 2\theta + 4\theta = 90^\circ \]

### Step 2: Simplify the equation
Combine the terms:
\[ 6\theta = 90^\circ \]

### Step 3: Solve for \( \theta \)
Now, divide both sides by 6:
\[ \theta = \frac{90^\circ}{6} = 15^\circ \]

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

### Step 5: Conclusion
The value of \( 3\theta \) is \( 45^\circ \).

### Final Answer
Thus, the value of \( 3\theta \) is \( 45^\circ \).

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