The general solution of the equation tan `3x + cot (2x + (pi)/(3)) = 0` is :

Options

A. `n pi + (pi)/(3), n in z`
B. `2 npi pm (pi)/(6), n in z`
C. `2 n pi pm (pi)/(3), n in z`
D. `n pi + (5pi)/(6), n in z`

Answer

To solve the equation \( \tan(3x) + \cot(2x + \frac{\pi}{3}) = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan(3x) + \cot(2x + \frac{\pi}{3}) = 0 \]
This can be rewritten as:
\[ \tan(3x) = -\cot(2x + \frac{\pi}{3}) \]

### Step 2: Use the cotangent identity
Recall that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Thus, we can rewrite the equation as:
\[ \tan(3x) = -\frac{1}{\tan(2x + \frac{\pi}{3})} \]
This implies:
\[ \tan(3x) \tan(2x + \frac{\pi}{3}) = -1 \]

### Step 3: Use the tangent addition formula
Using the tangent addition formula, we have:
\[ \tan(2x + \frac{\pi}{3}) = \frac{\tan(2x) + \tan(\frac{\pi}{3})}{1 - \tan(2x)\tan(\frac{\pi}{3})} \]
Since \( \tan(\frac{\pi}{3}) = \sqrt{3} \), we can substitute this into our equation:
\[ \tan(3x) \left( \frac{\tan(2x) + \sqrt{3}}{1 - \tan(2x)\sqrt{3}} \right) = -1 \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ \tan(3x)(\tan(2x) + \sqrt{3}) = - (1 - \tan(2x)\sqrt{3}) \]

### Step 5: Solve for angles
We can simplify and solve for the angles. However, we can also use the property of tangent functions. Since \( \tan(A) = \tan(B) \) implies \( A = B + n\pi \) for any integer \( n \), we can equate:
\[ 3x = \frac{5\pi}{6} + 2x + n\pi \]

### Step 6: Isolate \( x \)
Rearranging gives:
\[ 3x - 2x = \frac{5\pi}{6} + n\pi \]
\[ x = \frac{5\pi}{6} + n\pi \]

### Step 7: General solution
Thus, the general solution for the equation is:
\[ x = \frac{5\pi}{6} + n\pi, \quad n \in \mathbb{Z} \]

### Final Answer
The general solution of the equation \( \tan(3x) + \cot(2x + \frac{\pi}{3}) = 0 \) is:
\[ x = \frac{5\pi}{6} + n\pi, \quad n \in \mathbb{Z} \]