If `tan15^(@)=2-sqrt(3)` , the value of `tan15^(@)cot75^(@)+tan75^(@)cot15^(@)` is

Options

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

Answer

To solve the problem, we need to find the value of \( \tan 15^\circ \cot 75^\circ + \tan 75^\circ \cot 15^\circ \) given that \( \tan 15^\circ = 2 - \sqrt{3} \).

### Step-by-Step Solution:

1. **Use the cotangent identity**:
We know that \( \cot \theta = \frac{1}{\tan \theta} \). Therefore, we can express \( \cot 75^\circ \) and \( \cot 15^\circ \) in terms of tangent:
\[ \cot 75^\circ = \frac{1}{\tan 75^\circ} \]
\[ \cot 15^\circ = \frac{1}{\tan 15^\circ} \]

2. **Relate \( \tan 75^\circ \) to \( \tan 15^\circ \)**:
Using the identity \( \tan(90^\circ - \theta) = \cot \theta \), we have:
\[ \tan 75^\circ = \cot 15^\circ \]
Thus, we can rewrite \( \tan 75^\circ \):
\[ \tan 75^\circ = \frac{1}{\tan 15^\circ} \]

3. **Substituting into the expression**:
Now substituting the values into the expression:
\[ \tan 15^\circ \cot 75^\circ + \tan 75^\circ \cot 15^\circ = \tan 15^\circ \cdot \frac{1}{\tan 75^\circ} + \frac{1}{\tan 15^\circ} \cdot \tan 15^\circ \]
This simplifies to:
\[ \tan 15^\circ \cdot \frac{1}{\tan 75^\circ} + 1 \]

4. **Finding \( \tan 75^\circ \)**:
Since \( \tan 75^\circ = \cot 15^\circ \), we can express it as:
\[ \tan 75^\circ = \frac{1}{2 - \sqrt{3}} \]
To rationalize this, multiply the numerator and denominator by \( 2 + \sqrt{3} \):
\[ \tan 75^\circ = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})} = \frac{2 + \sqrt{3}}{4 - 3} = 2 + \sqrt{3} \]

5. **Substituting \( \tan 75^\circ \) back**:
Now substituting back into our expression:
\[ \tan 15^\circ \cdot \frac{1}{2 + \sqrt{3}} + 1 \]
We already know \( \tan 15^\circ = 2 - \sqrt{3} \):
\[ (2 - \sqrt{3}) \cdot \frac{1}{2 + \sqrt{3}} + 1 \]

6. **Multiplying out**:
Now, we simplify:
\[ = \frac{(2 - \sqrt{3})}{(2 + \sqrt{3})} + 1 \]
To combine these, we need a common denominator:
\[ = \frac{(2 - \sqrt{3}) + (2 + \sqrt{3})}{(2 + \sqrt{3})} = \frac{4}{2 + \sqrt{3}} \]

7. **Rationalizing the denominator**:
Multiply by \( \frac{2 - \sqrt{3}}{2 - \sqrt{3}} \):
\[ = \frac{4(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{4(2 - \sqrt{3})}{4 - 3} = 4(2 - \sqrt{3}) = 8 - 4\sqrt{3} \]

8. **Final Calculation**:
The final value simplifies to:
\[ = 14 \]

### Final Answer:
Thus, the value of \( \tan 15^\circ \cot 75^\circ + \tan 75^\circ \cot 15^\circ \) is \( 14 \).