Number of solution of `tan(2x)= tan(6x)` in `(0,3pi)` is :

Options

A. 4
B. 5
C. 3
D. None of these

Answer

To find the number of solutions for the equation \( \tan(2x) = \tan(6x) \) in the interval \( (0, 3\pi) \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan(2x) = \tan(6x) \]
This implies that:
\[ 2x = 6x + n\pi \quad \text{for some integer } n \]
This is because the tangent function is periodic with a period of \( \pi \).

### Step 2: Rearrange the equation
Rearranging gives:
\[ 2x - 6x = n\pi \implies -4x = n\pi \implies x = -\frac{n\pi}{4} \]

### Step 3: Determine the values of \( n \)
Now, we need to find values of \( n \) such that \( x \) lies in the interval \( (0, 3\pi) \):
\[ 0 < -\frac{n\pi}{4} < 3\pi \]
This can be split into two inequalities:
1. \( -\frac{n\pi}{4} > 0 \) (which implies \( n < 0 \))
2. \( -\frac{n\pi}{4} < 3\pi \) (which implies \( n > -12 \))

### Step 4: Find integer values of \( n \)
From the inequalities, we find that \( n \) can take integer values from \( -11 \) to \( -1 \). The possible integer values of \( n \) are:
\[ -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1 \]
This gives us a total of 11 integer values.

### Step 5: Count the solutions
Each integer value of \( n \) corresponds to a unique solution for \( x \):
\[ x = -\frac{n\pi}{4} \]
Thus, there are a total of 11 solutions in the interval \( (0, 3\pi) \).

### Conclusion
The number of solutions of \( \tan(2x) = \tan(6x) \) in the interval \( (0, 3\pi) \) is:
\[ \text{Answer: } 11 \]