The general solution of `(tan 5x-tan 4x)/(1+ tan 5 x tan 4x)=1` is

Options

A. `n pi+pi/4, AA n in Z`
B. `n pi pm pi/4, AA n in Z`
C. `phi`
D. `n pi=pi/6, AA n in Z`

Explanation

To solve the equation \(\frac{\tan 5x - \tan 4x}{1 + \tan 5x \tan 4x} = 1\), we can use the tangent subtraction formula.

### Step-by-Step Solution:

1. **Recognize the Identity**:
The expression \(\frac{\tan A - \tan B}{1 + \tan A \tan B} = \tan(A - B)\).
Here, let \(A = 5x\) and \(B = 4x\).

Therefore, we can rewrite the equation as:
\[ \tan(5x - 4x) = 1 \]

2. **Simplify the Left Side**:
This simplifies to:
\[ \tan(x) = 1 \]

3. **Find General Solutions**:
The general solution for \(\tan(x) = 1\) is:
\[ x = n\pi + \frac{\pi}{4} \quad \text{for } n \in \mathbb{Z} \]

4. **Substitute Back**:
Since we want the solution in terms of \(x\), we can express it as:
\[ x = n\pi + \frac{\pi}{4} \]

5. **Express in terms of \(x\)**:
Since we are looking for the general solution of the original equation, we can write:
\[ 5x - 4x = n\pi + \frac{\pi}{4} \]
This gives:
\[ x = n\pi + \frac{\pi}{4} \]

### Final Answer:
Thus, the general solution of the equation is:
\[ x = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \]