The value of `cotx+cot(60^@+x)+cot(120^@+x)` is equal to

Options

A. `cot 3x`
B. `tan3x`
C. `3tan 3x`
D. `(3-9tan^(2)x)/(3tanx-tan^(3)x)`

Answer

To solve the problem \( \cot x + \cot(60^\circ + x) + \cot(120^\circ + x) \), we will use the cotangent addition formulas and some properties of trigonometric functions.

### Step-by-Step Solution:

1. **Write the expression**:
\[ \cot x + \cot(60^\circ + x) + \cot(120^\circ + x) \]

2. **Use the cotangent addition formula**:
The cotangent addition formula states:
\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]
We will apply this to \( \cot(60^\circ + x) \) and \( \cot(120^\circ + x) \).

3. **Calculate \( \cot(60^\circ + x) \)**:
Using \( \cot(60^\circ) = \frac{1}{\sqrt{3}} \):
\[ \cot(60^\circ + x) = \frac{\cot 60^\circ \cot x - 1}{\cot 60^\circ + \cot x} = \frac{\frac{1}{\sqrt{3}} \cot x - 1}{\frac{1}{\sqrt{3}} + \cot x} \]

4. **Calculate \( \cot(120^\circ + x) \)**:
Using \( \cot(120^\circ) = -\frac{1}{\sqrt{3}} \):
\[ \cot(120^\circ + x) = \frac{\cot 120^\circ \cot x - 1}{\cot 120^\circ + \cot x} = \frac{-\frac{1}{\sqrt{3}} \cot x - 1}{-\frac{1}{\sqrt{3}} + \cot x} \]

5. **Combine the terms**:
Now substituting these back into the original expression:
\[ \cot x + \frac{\frac{1}{\sqrt{3}} \cot x - 1}{\frac{1}{\sqrt{3}} + \cot x} + \frac{-\frac{1}{\sqrt{3}} \cot x - 1}{-\frac{1}{\sqrt{3}} + \cot x} \]

6. **Find a common denominator**:
The common denominator for the two fractions is:
\[ \left(\frac{1}{\sqrt{3}} + \cot x\right)\left(-\frac{1}{\sqrt{3}} + \cot x\right) \]

7. **Simplify the expression**:
After simplifying and combining all terms, we will arrive at:
\[ \frac{3 - 9 \tan^2 x}{3 \tan x - \tan^3 x} \]

8. **Final result**:
Thus, the value of \( \cot x + \cot(60^\circ + x) + \cot(120^\circ + x) \) simplifies to:
\[ \frac{3 - 9 \tan^2 x}{3 \tan x - \tan^3 x} \]

### Conclusion:
The final answer is:
\[ \boxed{\frac{3 - 9 \tan^2 x}{3 \tan x - \tan^3 x}} \]