The value of `"tan"75^(@)-"cot"75^(@)` is equal to

Options

A. `2/sqrt(3)`
B. `2 * sqrt(3)`
C. `2-sqrt(3)`
D. 1

Answer

To solve the problem of finding the value of \( \tan 75^\circ - \cot 75^\circ \), we can follow these steps:

### Step 1: Express \( \tan 75^\circ \)
We can express \( \tan 75^\circ \) using the angle addition formula:
\[ \tan 75^\circ = \tan(45^\circ + 30^\circ) \]
Using the formula \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \), we have:
\[ \tan 75^\circ = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} \]
Where \( \tan 45^\circ = 1 \) and \( \tan 30^\circ = \frac{1}{\sqrt{3}} \).

### Step 2: Substitute the values
Substituting the values into the formula:
\[ \tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \]
This simplifies to:
\[ \tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} \]

### Step 3: Simplify the numerator and denominator
To simplify, we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ \tan 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Step 4: Express \( \cot 75^\circ \)
Now, we find \( \cot 75^\circ \):
\[ \cot 75^\circ = \frac{1}{\tan 75^\circ} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

### Step 5: Calculate \( \tan 75^\circ - \cot 75^\circ \)
Now we can compute \( \tan 75^\circ - \cot 75^\circ \):
\[ \tan 75^\circ - \cot 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} - \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

### Step 6: Find a common denominator
The common denominator is \( (\sqrt{3} - 1)(\sqrt{3} + 1) \):
\[ \tan 75^\circ - \cot 75^\circ = \frac{(\sqrt{3} + 1)^2 - (\sqrt{3} - 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \]

### Step 7: Expand the squares
Expanding the squares in the numerator:
\[ (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]
\[ (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \]
Thus,
\[ \tan 75^\circ - \cot 75^\circ = \frac{(4 + 2\sqrt{3}) - (4 - 2\sqrt{3})}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{4\sqrt{3}}{(\sqrt{3})^2 - 1^2} = \frac{4\sqrt{3}}{3 - 1} = \frac{4\sqrt{3}}{2} \]

### Step 8: Simplify the result
This simplifies to:
\[ \tan 75^\circ - \cot 75^\circ = 2\sqrt{3} \]

### Final Answer
Thus, the value of \( \tan 75^\circ - \cot 75^\circ \) is:
\[ \boxed{2\sqrt{3}} \]