If tan 2θ.tan 4θ = 1, then the value of tan 3θ is :

Question

Accepted Answer

To solve the problem, we need to find the value of $\tan 3\theta$ given that $\tan 2\theta \cdot \tan 4\theta = 1$.

### Explanation:

1. **Given Equation:**
   $$
   \tan 2\theta \cdot \tan 4\theta = 1
   $$

This implies that $\tan 2\theta$ and $\tan 4\theta$ are reciprocal to each other. Therefore, $\tan 4\theta = \frac{1}{\tan 2\theta}$.

2. **Using Trigonometric Identities:**
   We know the identity for tangent of sum:
   $$
   \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}
   $$

Applying this identity to $\tan 3\theta$, we can express it as:
   $$
   \tan 3\theta = \tan(2\theta + \theta) = \frac{\tan 2\theta + \tan \theta}{1 - \tan 2\theta \tan \theta}
   $$

3. **Using the Given Condition:**
   Since $\tan 4\theta = \frac{1}{\tan 2\theta}$, we can use the identity for $\tan 4\theta$ in terms of $\tan 2\theta$:
   $$
   \tan 4\theta = \frac{2\tan 2\theta}{1 - \tan^2 2\theta}
   $$

Setting this equal to $\frac{1}{\tan 2\theta}$, we have:
   $$
   \frac{2\tan 2\theta}{1 - \tan^2 2\theta} = \frac{1}{\tan 2\theta}
   $$

Solving this equation, we find that $\tan 2\theta = 1$.

4. **Finding $\tan 3\theta$:**
   Substitute $\tan 2\theta = 1$ into the expression for $\tan 3\theta$:
   $$
   \tan 3\theta = \frac{1 + \tan \theta}{1 - 1 \cdot \tan \theta} = \frac{1 + \tan \theta}{1 - \tan \theta}
   $$

If $\tan \theta = 0$, then:
   $$
   \tan 3\theta = \frac{1 + 0}{1 - 0} = 1
   $$

Therefore, the value of $\tan 3\theta$ is indeed 1.

### Conclusion:

- **Option 2 (1) is correct** because the calculations show that $\tan 3\theta = 1$ when $\tan 2\theta = 1$.

- **Why Other Options are Incorrect:**
  - **Option 1 (0):** This would imply $\tan 3\theta = 0$, which is not possible given the calculations.
  - **Option 3 ($\sqrt{3}$):** This value does not satisfy the derived expression for $\tan 3\theta$.
  - **Option 4 (∞):** This would imply $\tan 3\theta$ is undefined, which contradicts the derived value of 1.

Thus, the correct answer is Option 2, $\tan 3\theta = 1$.