`cot theta + cot (theta+(pi)/(3)) + cot (theta + (2 pi)/(3))` is equal to

Options

A. `- cot 3 theta`
B. `cot 3 theta`
C. `2 cot 3 theta`
D. `3 cot 3 theta`

Answer

To solve the expression \( \cot \theta + \cot \left( \theta + \frac{\pi}{3} \right) + \cot \left( \theta + \frac{2\pi}{3} \right) \), we will follow these steps:

### Step 1: Rewrite the cotangent terms
We know that:
- \( \cot \left( \theta + \frac{2\pi}{3} \right) = \cot \left( \pi - \left( \frac{\pi}{3} + \theta \right) \right) = -\cot \left( \frac{\pi}{3} + \theta \right) \)

So we can rewrite the expression as:
\[ \cot \theta + \cot \left( \theta + \frac{\pi}{3} \right) - \cot \left( \frac{\pi}{3} + \theta \right) \]

### Step 2: Use the cotangent addition formula
Using the cotangent addition formula:
\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]
Let \( a = \theta \) and \( b = \frac{\pi}{3} \):
\[ \cot \left( \theta + \frac{\pi}{3} \right) = \frac{\cot \theta \cot \frac{\pi}{3} - 1}{\cot \theta + \cot \frac{\pi}{3}} \]
Since \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \), we have:
\[ \cot \left( \theta + \frac{\pi}{3} \right) = \frac{\cot \theta \cdot \frac{1}{\sqrt{3}} - 1}{\cot \theta + \frac{1}{\sqrt{3}}} \]

### Step 3: Substitute and simplify
Now substituting back into our expression:
\[ \cot \theta + \frac{\cot \theta \cdot \frac{1}{\sqrt{3}} - 1}{\cot \theta + \frac{1}{\sqrt{3}}} - \frac{\cot \theta \cdot \frac{1}{\sqrt{3}} - 1}{\cot \theta + \frac{1}{\sqrt{3}}} \]
Notice that the last two terms cancel out. Thus, we are left with:
\[ \cot \theta \]

### Step 4: Final result
Thus, the expression simplifies to:
\[ \cot \theta \]

### Summary
The value of \( \cot \theta + \cot \left( \theta + \frac{\pi}{3} \right) + \cot \left( \theta + \frac{2\pi}{3} \right) \) is equal to \( \cot \theta \).