If `tan15^@=2-sqrt3`, then the value of `cot^2""75^@`

Options

A. `7+sqrt3`
B. `7-2sqrt3`
C. `7-4sqrt3`
D. `7+4sqrt3`

Answer

To find the value of \( \cot^2 75^\circ \) given that \( \tan 15^\circ = 2 - \sqrt{3} \), we can follow these steps:

### Step 1: Use the cotangent identity
We know that:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
Thus, we can express \( \cot 75^\circ \) in terms of \( \tan 15^\circ \):
\[ \cot 75^\circ = \tan(90^\circ - 75^\circ) = \tan 15^\circ \]

### Step 2: Substitute the given value
From the problem, we have:
\[ \tan 15^\circ = 2 - \sqrt{3} \]
So,
\[ \cot 75^\circ = 2 - \sqrt{3} \]

### Step 3: Square the cotangent value
Now, we need to find \( \cot^2 75^\circ \):
\[ \cot^2 75^\circ = (2 - \sqrt{3})^2 \]

### Step 4: Expand the square
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ \cot^2 75^\circ = (2)^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
Calculating each term:
\[ = 4 - 4\sqrt{3} + 3 \]

### Step 5: Combine the terms
Now, combine the constant terms:
\[ \cot^2 75^\circ = 7 - 4\sqrt{3} \]

Thus, the final answer is:
\[ \cot^2 75^\circ = 7 - 4\sqrt{3} \]

---