The general solution of the equation `"tan" 3x = "tan" 5x`, is

Options

A. `x = (npi)/(2), n in Z`
B. `x = n pi, n in Z`
C. `x = (2n+1) pi, n in Z`
D. none of these

Answer

To solve the equation \( \tan(3x) = \tan(5x) \), we can use the property of the tangent function. The general solution for the equation \( \tan(\theta_1) = \tan(\theta_2) \) is given by:

\[ \theta_1 = \theta_2 + n\pi \]

where \( n \) is any integer.

### Step-by-step Solution:

1. **Set up the equation:**
We start with the equation:
\[ \tan(3x) = \tan(5x) \]

2. **Apply the property of tangent:**
Using the property of tangent, we can write:
\[ 3x = 5x + n\pi \]
where \( n \) is any integer.

3. **Rearrange the equation:**
Rearranging the equation gives:
\[ 3x - 5x = n\pi \]
This simplifies to:
\[ -2x = n\pi \]

4. **Solve for \( x \):**
Dividing both sides by -2, we find:
\[ x = -\frac{n\pi}{2} \]

5. **Express the solution in a standard form:**
We can rewrite this as:
\[ x = \frac{n\pi}{2} \]
where \( n \) can be any integer (positive, negative, or zero).

### Final General Solution:
Thus, the general solution of the equation \( \tan(3x) = \tan(5x) \) is:
\[ x = \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]