Number of solution of ` tan(2x)= tan(6x) ` in ` ( 0, 3 pi)` 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: Use the periodic property of tangent
Since \( \tan(A) = \tan(B) \) implies \( A = B + n\pi \) for some integer \( n \), we can write:
\[ 2x = 6x + n\pi \]

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

### Step 3: Determine valid \( n \) values
We need to find the values of \( n \) such that \( x \) lies within 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: Identify integer values for \( n \)
The integer values of \( n \) that satisfy \( -12 < n < 0 \) are:
\[ n = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1 \]
This gives us a total of 11 possible values for \( n \).

### Step 5: Calculate the corresponding \( x \) values
For each valid \( n \):
- When \( n = -1 \), \( x = \frac{\pi}{4} \)
- When \( n = -2 \), \( x = \frac{\pi}{2} \)
- When \( n = -3 \), \( x = \frac{3\pi}{4} \)
- When \( n = -4 \), \( x = \pi \)
- When \( n = -5 \), \( x = \frac{5\pi}{4} \)
- When \( n = -6 \), \( x = \frac{3\pi}{2} \)
- When \( n = -7 \), \( x = \frac{7\pi}{4} \)
- When \( n = -8 \), \( x = 2\pi \)
- When \( n = -9 \), \( x = \frac{9\pi}{4} \)
- When \( n = -10 \), \( x = \frac{5\pi}{2} \)
- When \( n = -11 \), \( x = \frac{11\pi}{4} \)

### Step 6: Count valid solutions
Now we need to check which of these values lie within \( (0, 3\pi) \):
- \( \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}, 2\pi, \frac{9\pi}{4}, \frac{5\pi}{2}, \frac{11\pi}{4} \)

The values \( \frac{9\pi}{4}, \frac{5\pi}{2}, \frac{11\pi}{4} \) exceed \( 3\pi \), so we exclude them.

### Final Count
Thus, the valid solutions are:
1. \( \frac{\pi}{4} \)
2. \( \frac{\pi}{2} \)
3. \( \frac{3\pi}{4} \)
4. \( \pi \)
5. \( \frac{5\pi}{4} \)
6. \( \frac{3\pi}{2} \)
7. \( \frac{7\pi}{4} \)
8. \( 2\pi \)

This gives us a total of 8 solutions in the interval \( (0, 3\pi) \).

### Conclusion
The number of solutions of \( \tan(2x) = \tan(6x) \) in \( (0, 3\pi) \) is **8**.

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