`Tan 75 - Cot 75=`

Options

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

Explanation

To solve the expression \( \tan 75^\circ - \cot 75^\circ \), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We know that:
\[ \tan 75^\circ = \frac{\sin 75^\circ}{\cos 75^\circ} \quad \text{and} \quad \cot 75^\circ = \frac{\cos 75^\circ}{\sin 75^\circ} \]
Thus, we can rewrite the expression:
\[ \tan 75^\circ - \cot 75^\circ = \frac{\sin 75^\circ}{\cos 75^\circ} - \frac{\cos 75^\circ}{\sin 75^\circ} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \( \cos 75^\circ \sin 75^\circ \). Therefore, we can combine the fractions:
\[ \tan 75^\circ - \cot 75^\circ = \frac{\sin^2 75^\circ - \cos^2 75^\circ}{\cos 75^\circ \sin 75^\circ} \]

### Step 3: Use the identity for sine and cosine
We can use the identity \( \sin^2 \theta - \cos^2 \theta = -\cos 2\theta \):
\[ \sin^2 75^\circ - \cos^2 75^\circ = -\cos(2 \times 75^\circ) = -\cos 150^\circ \]
Thus, we can rewrite the expression:
\[ \tan 75^\circ - \cot 75^\circ = \frac{-\cos 150^\circ}{\cos 75^\circ \sin 75^\circ} \]

### Step 4: Evaluate \( \cos 150^\circ \)
We know that:
\[ \cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2} \]
Substituting this back into our expression gives:
\[ \tan 75^\circ - \cot 75^\circ = \frac{-(-\frac{\sqrt{3}}{2})}{\cos 75^\circ \sin 75^\circ} = \frac{\frac{\sqrt{3}}{2}}{\cos 75^\circ \sin 75^\circ} \]

### Step 5: Use the double angle identity
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ \cos 75^\circ \sin 75^\circ = \frac{1}{2} \sin 150^\circ = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]
Thus, we can rewrite:
\[ \tan 75^\circ - \cot 75^\circ = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{4}} = \frac{\sqrt{3}}{2} \cdot 4 = 2\sqrt{3} \]

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