Given that `tantheta=mne0, tan2 theta=n ne0and tantheta+tan2theta=tan3theta`,

Options

A. m = n
B. m + n = 1
C. m + n = 0
D. mn =- 1

Explanation

To solve the problem step by step, we will start from the given equations and use trigonometric identities to find the relationship between \( m \) and \( n \).

### Step 1: Write down the given equations
We are given:
1. \( \tan \theta = m \) (where \( m \neq 0 \))
2. \( \tan 2\theta = n \) (where \( n \neq 0 \))
3. \( \tan \theta + \tan 2\theta = \tan 3\theta \)

### Step 2: Use the identity for \( \tan 3\theta \)
Using the formula for \( \tan 3\theta \):
\[ \tan 3\theta = \frac{\tan \theta + \tan 2\theta}{1 - \tan \theta \tan 2\theta} \]
Substituting the values of \( \tan \theta \) and \( \tan 2\theta \):
\[ \tan 3\theta = \frac{m + n}{1 - mn} \]

### Step 3: Set up the equation
Since we know from the problem statement that:
\[ \tan \theta + \tan 2\theta = \tan 3\theta \]
We can equate:
\[ m + n = \frac{m + n}{1 - mn} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ (m + n)(1 - mn) = m + n \]

### Step 5: Simplify the equation
Expanding the left-hand side:
\[ m + n - mn(m + n) = m + n \]
Subtract \( m + n \) from both sides:
\[ -mn(m + n) = 0 \]

### Step 6: Factor the equation
This gives us two possible cases:
1. \( mn = 0 \)
2. \( m + n = 0 \)

### Step 7: Analyze the cases
From \( mn = 0 \), either \( m = 0 \) or \( n = 0 \). However, since both \( m \) and \( n \) are given to be non-zero, this case does not apply.

From \( m + n = 0 \), we conclude:
\[ m + n = 0 \implies n = -m \]

### Final Result
Thus, the relationship between \( m \) and \( n \) is:
\[ m + n = 0 \]