The number of solution of the equation `tan3x-tan2x-tan3x.tan2x = 1` in `[0, 2pi]` is

Options

A. 1
B. Zero
C. 3
D. 2

Answer

To solve the equation \( \tan 3x - \tan 2x - \tan 3x \tan 2x = 1 \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step 1: Rearrange the Equation
We start with the equation:
\[ \tan 3x - \tan 2x - \tan 3x \tan 2x = 1 \]
Rearranging gives:
\[ \tan 3x - \tan 2x = 1 + \tan 3x \tan 2x \]

### Step 2: Use the Tangent Addition Formula
We can recognize that the left-hand side can be expressed using the tangent addition formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Thus, we can rewrite our equation as:
\[ \tan(3x - 2x) = \tan(1) \]
This simplifies to:
\[ \tan x = 1 \]

### Step 3: Solve for \( x \)
The general solution for \( \tan x = 1 \) is:
\[ x = \frac{\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z} \]
Now, we need to find the specific solutions in the interval \( [0, 2\pi] \).

### Step 4: Find Specific Solutions
1. For \( n = 0 \):
\[ x = \frac{\pi}{4} \]
2. For \( n = 1 \):
\[ x = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \]

Both solutions \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \) lie within the interval \( [0, 2\pi] \).

### Step 5: Count the Solutions
Thus, the total number of solutions in the interval \( [0, 2\pi] \) is:
\[ \text{Number of solutions} = 2 \]

### Final Answer
The number of solutions of the equation \( \tan 3x - \tan 2x - \tan 3x \tan 2x = 1 \) in the interval \( [0, 2\pi] \) is **2**.

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