If `tan 3 theta + tan theta =2 tan 2 theta`, then `theta` is equal to `(n in Z)`

Options

A. `n pi`
B. `(n pi)/4`
C. `(n pi)/2`
D. none of these

Explanation

To solve the equation \( \tan 3\theta + \tan \theta = 2 \tan 2\theta \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \tan 3\theta + \tan \theta = 2 \tan 2\theta \]
We can rearrange this to:
\[ \tan 3\theta - 2 \tan 2\theta + \tan \theta = 0 \]

### Step 2: Using the Tangent Addition Formula
Using the tangent addition formula, we can express \( \tan 3\theta \) as:
\[ \tan 3\theta = \tan(2\theta + \theta) = \frac{\tan 2\theta + \tan \theta}{1 - \tan 2\theta \tan \theta} \]
Substituting this into our equation gives:
\[ \frac{\tan 2\theta + \tan \theta}{1 - \tan 2\theta \tan \theta} - 2 \tan 2\theta + \tan \theta = 0 \]

### Step 3: Simplifying the Equation
Multiplying through by \( 1 - \tan 2\theta \tan \theta \) to eliminate the denominator:
\[ \tan 2\theta + \tan \theta - 2 \tan 2\theta (1 - \tan 2\theta \tan \theta) + \tan \theta (1 - \tan 2\theta \tan \theta) = 0 \]
This simplifies to:
\[ \tan 2\theta + \tan \theta - 2 \tan 2\theta + 2 \tan 2\theta \tan \theta \tan \theta + \tan \theta - \tan \theta \tan 2\theta \tan \theta = 0 \]

### Step 4: Collecting Like Terms
Rearranging the terms gives us:
\[ -\tan 2\theta + \tan \theta (1 + \tan 2\theta \tan \theta) = 0 \]

### Step 5: Setting Each Factor to Zero
This leads to two cases:
1. \( \tan 2\theta = 0 \)
2. \( 1 + \tan 2\theta \tan \theta = 0 \)

### Step 6: Solving \( \tan 2\theta = 0 \)
For \( \tan 2\theta = 0 \):
\[ 2\theta = n\pi \quad \Rightarrow \quad \theta = \frac{n\pi}{2} \]
where \( n \in \mathbb{Z} \).

### Step 7: Solving \( 1 + \tan 2\theta \tan \theta = 0 \)
For \( 1 + \tan 2\theta \tan \theta = 0 \):
\[ \tan 2\theta \tan \theta = -1 \]
This case can lead to more complex solutions, but for simplicity, we will focus on the first case.

### Conclusion
Thus, the solution for \( \theta \) is:
\[ \theta = \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]