If ` tan alpha + cot alpha =7` . What is the value of tan ` alpha - cot alpha ` ?

Options

A. ` 2 sqrt""5 `
B. ` sqrt"" 5`
C. ` 3 sqrt"" 5`
D. None of these

Answer

To solve the equation \( \tan \alpha + \cot \alpha = 7 \) and find the value of \( \tan \alpha - \cot \alpha \), we can follow these steps:

### Step 1: Define Variables
Let \( x = \tan \alpha \). Then, we have:
\[ \cot \alpha = \frac{1}{\tan \alpha} = \frac{1}{x} \]
Thus, the equation becomes:
\[ x + \frac{1}{x} = 7 \]

### Step 2: Multiply by \( x \)
To eliminate the fraction, multiply both sides by \( x \):
\[ x^2 + 1 = 7x \]

### Step 3: Rearrange the Equation
Rearranging gives us a standard quadratic equation:
\[ x^2 - 7x + 1 = 0 \]

### Step 4: Use the Quadratic Formula
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -7, c = 1 \):
\[ x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{7 \pm \sqrt{49 - 4}}{2} \]
\[ x = \frac{7 \pm \sqrt{45}}{2} \]
\[ x = \frac{7 \pm 3\sqrt{5}}{2} \]

### Step 5: Find \( \tan \alpha - \cot \alpha \)
Now, we need to find \( \tan \alpha - \cot \alpha \):
\[ \tan \alpha - \cot \alpha = x - \frac{1}{x} \]
Using the identity \( (x + \frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2} \):
\[ (x - \frac{1}{x})^2 = (x + \frac{1}{x})^2 - 4 \]
Substituting \( x + \frac{1}{x} = 7 \):
\[ (x - \frac{1}{x})^2 = 7^2 - 4 = 49 - 4 = 45 \]
Taking the square root:
\[ x - \frac{1}{x} = \sqrt{45} = 3\sqrt{5} \]

### Final Answer
Thus, the value of \( \tan \alpha - \cot \alpha \) is:
\[ \boxed{3\sqrt{5}} \]