The solution set of the equation `tanax = tanbx`

Options

A. An A.P. with common difference `pi/(a-b)` or `pi/(a+b)`
B. An H.P with common `pi/(|a-b|)`
C. An A.P. with common difference `pi/(a+b)`
D. An A.P. with common difference `pi/(a-b)` or `pi/(b-a)`

Answer

To solve the equation \( \tan(Ax) = \tan(Bx) \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the Equation**:
\[ \tan(Ax) = \tan(Bx) \]

2. **Use the Property of Tangent**:
The property of the tangent function states that if \( \tan(\theta_1) = \tan(\theta_2) \), then:
\[ \theta_1 = \theta_2 + n\pi \quad \text{for } n \in \mathbb{Z} \]
Applying this to our equation, we have:
\[ Ax = Bx + n\pi \]

3. **Rearrange the Equation**:
Move the \( Bx \) term to the left side:
\[ Ax - Bx = n\pi \]
This simplifies to:
\[ (A - B)x = n\pi \]

4. **Solve for \( x \)**:
Divide both sides by \( (A - B) \) (assuming \( A \neq B \)):
\[ x = \frac{n\pi}{A - B} \]

5. **Identify the Solution Set**:
Since \( n \) can take any integer value, the solution set can be expressed as:
\[ x = \frac{n\pi}{A - B}, \quad n \in \mathbb{Z} \]

### Final Solution:
The solution set of the equation \( \tan(Ax) = \tan(Bx) \) is:
\[ x = \frac{n\pi}{A - B}, \quad n \in \mathbb{Z} \]