If `tan 15^@=2-sqrt3`, then the value of `tan15^@ cot 75^@+tan75^@ cot 15^@` is

Options

A. 14
B. 12
C. 10
D. 8

Answer

To solve the problem, we need to find the value of \( \tan 15^\circ \cot 75^\circ + \tan 75^\circ \cot 15^\circ \).

### Step-by-Step Solution:

1. **Identify the given value**: We know that \( \tan 15^\circ = 2 - \sqrt{3} \).

2. **Use cotangent identities**:
- We know that \( \cot 75^\circ = \tan(90^\circ - 75^\circ) = \tan 15^\circ \).
- Similarly, \( \tan 75^\circ = \cot(90^\circ - 15^\circ) = \cot 15^\circ \).

3. **Substituting values**:
- Therefore, we can rewrite the expression as:
\[ \tan 15^\circ \cot 75^\circ + \tan 75^\circ \cot 15^\circ = \tan 15^\circ \tan 15^\circ + \cot 15^\circ \cot 15^\circ \]
- This simplifies to:
\[ \tan^2 15^\circ + \cot^2 15^\circ \]

4. **Using the identity for cotangent**:
- Recall that \( \cot 15^\circ = \frac{1}{\tan 15^\circ} \).
- Therefore, \( \cot^2 15^\circ = \frac{1}{\tan^2 15^\circ} \).

5. **Let \( x = \tan 15^\circ \)**:
- Then we have:
\[ \tan^2 15^\circ + \cot^2 15^\circ = x^2 + \frac{1}{x^2} \]

6. **Using the identity**:
- We can use the identity \( x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \).
- First, we need to find \( x + \frac{1}{x} \):
\[ x = 2 - \sqrt{3} \]
\[ \frac{1}{x} = \frac{1}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})} = \frac{2 + \sqrt{3}}{4 - 3} = 2 + \sqrt{3} \]
- Therefore:
\[ x + \frac{1}{x} = (2 - \sqrt{3}) + (2 + \sqrt{3}) = 4 \]

7. **Calculating \( x^2 + \frac{1}{x^2} \)**:
- Now substitute back into the identity:
\[ x^2 + \frac{1}{x^2} = (4)^2 - 2 = 16 - 2 = 14 \]

8. **Final answer**:
- Thus, the value of \( \tan 15^\circ \cot 75^\circ + \tan 75^\circ \cot 15^\circ \) is \( 14 \).