If `tan theta=2-sqrt3`, then `tan(90-theta)` is equal to :

Options

A. `2+sqrt3`
B. `2-sqrt3`
C. `3+sqrt2`
D. `3-sqrt2`

Answer

To solve the problem, we need to find the value of \( \tan(90^\circ - \theta) \) given that \( \tan \theta = 2 - \sqrt{3} \).

### Step-by-Step Solution:

1. **Use the Co-Function Identity**:
We know that:
\[ \tan(90^\circ - \theta) = \cot \theta \]

2. **Find the Value of \( \cot \theta \)**:
The cotangent is the reciprocal of the tangent:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Given \( \tan \theta = 2 - \sqrt{3} \), we can substitute this into the equation:
\[ \cot \theta = \frac{1}{2 - \sqrt{3}} \]

3. **Rationalize the Denominator**:
To simplify \( \frac{1}{2 - \sqrt{3}} \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 2 + \sqrt{3} \):
\[ \cot \theta = \frac{1 \cdot (2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]

4. **Calculate the Denominator**:
Using the difference of squares:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

5. **Final Expression**:
Thus, we have:
\[ \cot \theta = 2 + \sqrt{3} \]

6. **Conclusion**:
Therefore, the value of \( \tan(90^\circ - \theta) \) is:
\[ \tan(90^\circ - \theta) = 2 + \sqrt{3} \]

### Final Answer:
\[ \tan(90^\circ - \theta) = 2 + \sqrt{3} \]