For `m ne n`, if `"tan" m theta = "tan" n theta`, then different values of `theta` are in

Options

A. A.P
B. H.P
C. G.P
D. no particular sequence

Answer

To solve the equation \( \tan(m\theta) = \tan(n\theta) \) for \( m \neq n \), we can follow these steps:

### Step 1: Understand the condition
The equation \( \tan(m\theta) = \tan(n\theta) \) implies that the angles \( m\theta \) and \( n\theta \) differ by an integer multiple of \( \pi \). This can be expressed as:
\[ m\theta = n\theta + k\pi \quad \text{for some integer } k \]

### Step 2: Rearranging the equation
Rearranging the equation gives us:
\[ m\theta - n\theta = k\pi \]
This simplifies to:
\[ (m - n)\theta = k\pi \]

### Step 3: Solve for \( \theta \)
Now, we can isolate \( \theta \):
\[ \theta = \frac{k\pi}{m - n} \]

### Step 4: General solution
Since \( k \) can be any integer, we can express the general solution for \( \theta \) as:
\[ \theta = \frac{k\pi}{m - n} \quad \text{where } k \in \mathbb{Z} \]

### Step 5: Identify the sequence of solutions
The values of \( \theta \) will form an arithmetic progression (AP) with a common difference of:
\[ \frac{\pi}{m - n} \]
Thus, the different values of \( \theta \) are:
\[ \theta = \frac{\pi}{m - n}, \frac{2\pi}{m - n}, \frac{3\pi}{m - n}, \ldots \]

### Conclusion
The different values of \( \theta \) are in an arithmetic progression with a common difference of \( \frac{\pi}{m - n} \).

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