If `tan alpha = sqrt3+2`, then the value of `tan alpha - cot alpha` is

Options

A. 2
B. `2sqrt3`
C. `sqrt3-2`
D. 4

Answer

To solve the problem, we need to find the value of \( \tan \alpha - \cot \alpha \) given that \( \tan \alpha = \sqrt{3} + 2 \).

### Step-by-step Solution:

1. **Identify the given value**:
\[ \tan \alpha = \sqrt{3} + 2 \]

2. **Calculate \( \cot \alpha \)**:
\[ \cot \alpha = \frac{1}{\tan \alpha} = \frac{1}{\sqrt{3} + 2} \]

3. **Rationalize the denominator of \( \cot \alpha \)**:
To rationalize, multiply the numerator and denominator by the conjugate of the denominator:
\[ \cot \alpha = \frac{1}{\sqrt{3} + 2} \cdot \frac{\sqrt{3} - 2}{\sqrt{3} - 2} = \frac{\sqrt{3} - 2}{(\sqrt{3} + 2)(\sqrt{3} - 2)} \]

4. **Calculate the denominator using the difference of squares**:
\[ (\sqrt{3} + 2)(\sqrt{3} - 2) = (\sqrt{3})^2 - (2)^2 = 3 - 4 = -1 \]
So, we have:
\[ \cot \alpha = \frac{\sqrt{3} - 2}{-1} = 2 - \sqrt{3} \]

5. **Now calculate \( \tan \alpha - \cot \alpha \)**:
\[ \tan \alpha - \cot \alpha = (\sqrt{3} + 2) - (2 - \sqrt{3}) \]
Simplifying this:
\[ = \sqrt{3} + 2 - 2 + \sqrt{3} = 2\sqrt{3} \]

6. **Final result**:
\[ \tan \alpha - \cot \alpha = 2\sqrt{3} \]

### Conclusion:
The value of \( \tan \alpha - \cot \alpha \) is \( 2\sqrt{3} \).