Find the value of `cot3theta`:

Options

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

Answer

To find the value of \( \cot 3\theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the definition of cotangent**:
\[ \cot 3\theta = \frac{1}{\tan 3\theta} \]
**Hint**: Remember that cotangent is the reciprocal of tangent.

2. **Express \( \tan 3\theta \)** using the angle addition formula:
\[ \tan 3\theta = \tan(2\theta + \theta) = \frac{\tan 2\theta + \tan \theta}{1 - \tan 2\theta \tan \theta} \]
**Hint**: Use the formula for \( \tan(a + b) \) to combine angles.

3. **Substitute \( \tan 2\theta \)** using the double angle formula:
\[ \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \]
**Hint**: This formula helps to express \( \tan 2\theta \) in terms of \( \tan \theta \).

4. **Substitute \( \tan 2\theta \) into the equation for \( \tan 3\theta \)**:
\[ \tan 3\theta = \frac{\frac{2\tan \theta}{1 - \tan^2 \theta} + \tan \theta}{1 - \frac{2\tan \theta}{1 - \tan^2 \theta} \tan \theta} \]
**Hint**: Combine the fractions carefully, ensuring you maintain the correct denominators.

5. **Simplify the numerator**:
\[ \tan 3\theta = \frac{2\tan \theta + \tan \theta(1 - \tan^2 \theta)}{1 - \frac{2\tan^2 \theta}{1 - \tan^2 \theta}} \]
**Hint**: Factor out \( \tan \theta \) from the numerator.

6. **Simplify the denominator**:
\[ \tan 3\theta = \frac{(2 + 1 - \tan^2 \theta)\tan \theta}{\frac{1 - \tan^2 \theta - 2\tan^2 \theta}{1 - \tan^2 \theta}} = \frac{(3 - \tan^2 \theta)\tan \theta}{\frac{1 - 3\tan^2 \theta}{1 - \tan^2 \theta}} \]
**Hint**: Ensure you simplify the fraction correctly, combining like terms.

7. **Final expression for \( \tan 3\theta \)**:
\[ \tan 3\theta = \frac{(3 - \tan^2 \theta)\tan \theta (1 - \tan^2 \theta)}{1 - 3\tan^2 \theta} \]
**Hint**: Keep track of all terms when simplifying.

8. **Find \( \cot 3\theta \)**:
\[ \cot 3\theta = \frac{1 - 3\tan^2 \theta}{(3 - \tan^2 \theta)\tan \theta (1 - \tan^2 \theta)} \]
**Hint**: Remember that \( \cot \) is the reciprocal of \( \tan \).

9. **Convert \( \tan \theta \) to \( \cot \theta \)**:
\[ \cot 3\theta = \frac{1 - 3 \cot^2 \theta}{3 \cot \theta - \cot^3 \theta} \]
**Hint**: Use the relationship \( \tan \theta = \frac{1}{\cot \theta} \) to make the conversion.

10. **Final Result**:
\[ \cot 3\theta = \frac{1 - 3 \cot^2 \theta}{3 \cot \theta - \cot^3 \theta} \]
**Hint**: This is the final expression for \( \cot 3\theta \).