If `tan a = sqrt(5) - 2`, then the value of `tan a - cot a = ? `

Options

A. `- 2`
B. `sqrt(5) + 2`
C. `-4`
D. `2 sqrt(5)`

Answer

To solve the problem where \( \tan a = \sqrt{5} - 2 \) and we need to find the value of \( \tan a - \cot a \), we can follow these steps:

### Step 1: Write down the given value of \( \tan a \).
We know that:
\[ \tan a = \sqrt{5} - 2 \]

### Step 2: Find the value of \( \cot a \).
Recall that:
\[ \cot a = \frac{1}{\tan a} \]
Substituting the value of \( \tan a \):
\[ \cot a = \frac{1}{\sqrt{5} - 2} \]

### Step 3: Rationalize \( \cot a \).
To rationalize \( \cot a \), we multiply the numerator and the denominator by the conjugate of the denominator:
\[ \cot a = \frac{1}{\sqrt{5} - 2} \cdot \frac{\sqrt{5} + 2}{\sqrt{5} + 2} \]
This gives us:
\[ \cot a = \frac{\sqrt{5} + 2}{(\sqrt{5} - 2)(\sqrt{5} + 2)} \]

### Step 4: Simplify the denominator.
Calculating the denominator:
\[ (\sqrt{5} - 2)(\sqrt{5} + 2) = 5 - 4 = 1 \]
Thus, we have:
\[ \cot a = \sqrt{5} + 2 \]

### Step 5: Calculate \( \tan a - \cot a \).
Now we can find \( \tan a - \cot a \):
\[ \tan a - \cot a = (\sqrt{5} - 2) - (\sqrt{5} + 2) \]
This simplifies to:
\[ \tan a - \cot a = \sqrt{5} - 2 - \sqrt{5} - 2 = -4 \]

### Final Answer:
\[ \tan a - \cot a = -4 \]