If tanx = `sqrt3` + 2, then the value of tanx - cotx is

Options

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

Answer

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

### Step-by-step Solution:

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

2. **Express cotangent in terms of tangent**:
\[ \cot x = \frac{1}{\tan x} = \frac{1}{\sqrt{3} + 2} \]

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

4. **Calculate the denominator**:
Using the difference of squares:
\[ (\sqrt{3} + 2)(\sqrt{3} - 2) = 3 - 4 = -1 \]
Thus,
\[ \cot x = \frac{\sqrt{3} - 2}{-1} = 2 - \sqrt{3} \]

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

6. **Final answer**:
Therefore, the value of \( \tan x - \cot x \) is:
\[ \tan x - \cot x = 2\sqrt{3} \]