Find the value of of `cot3theta:`

Options

A. `(cot^(3)theta-3cot theta)/(3cot^(2)theta)`
B. `(cot^(3)theta-3cot theta)/(3cot^(2)theta-1)`
C. `(3cot theta-cot^(3)theta)/(3cot^(2)theta-1)`
D. none of these

Answer

To find the value of \( \cot 3\theta \), we will use the formula for \( \tan 3\theta \) and then find \( \cot 3\theta \) as the reciprocal of \( \tan 3\theta \).

### Step-by-step Solution:

1. **Recall the formula for \( \tan 3\theta \)**:
\[ \tan 3\theta = \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta} \]

2. **Express \( \cot 3\theta \)**:
Since \( \cot 3\theta = \frac{1}{\tan 3\theta} \), we can substitute the formula for \( \tan 3\theta \):
\[ \cot 3\theta = \frac{1 - 3\tan^2 \theta}{3\tan \theta - \tan^3 \theta} \]

3. **Simplify the expression**:
We can rewrite \( \cot 3\theta \):
\[ \cot 3\theta = \frac{1 - 3\tan^2 \theta}{\tan \theta (3 - \tan^2 \theta)} \]

4. **Substituting \( \tan \theta \) with \( x \)** (where \( x = \tan \theta \)):
\[ \cot 3\theta = \frac{1 - 3x^2}{x(3 - x^2)} \]

5. **Final expression**:
Thus, we can express \( \cot 3\theta \) as:
\[ \cot 3\theta = \frac{1 - 3\tan^2 \theta}{3\tan \theta - \tan^3 \theta} \]

6. **Identify the correct option**:
From the options provided, we can see that the expression matches one of the options given in the question.

### Conclusion:
The value of \( \cot 3\theta \) is given by:
\[ \cot 3\theta = \frac{1 - 3\tan^2 \theta}{3\tan \theta - \tan^3 \theta} \]