To simplify the given expression [tex]\(\frac{1}{\sqrt{5}-2} - \frac{1}{\sqrt{5}+2}\)[/tex], we follow these steps:
### Step 1: Rationalizing the Denominators
To rationalize the denominators, we multiply the numerator and denominator by the conjugate of the denominator.
#### Rationalizing [tex]\(\frac{1}{\sqrt{5}-2}\)[/tex]
The conjugate of [tex]\(\sqrt{5} - 2\)[/tex] is [tex]\(\sqrt{5} + 2\)[/tex]. So,
[tex]\[
\frac{1}{\sqrt{5} - 2} \cdot \frac{\sqrt{5} + 2}{\sqrt{5} + 2} = \frac{\sqrt{5} + 2}{(\sqrt{5})^2 - (2)^2} = \frac{\sqrt{5} + 2}{5 - 4} = \frac{\sqrt{5} + 2}{1} = \sqrt{5} + 2
\][/tex]
#### Rationalizing [tex]\(\frac{1}{\sqrt{5}+2}\)[/tex]
The conjugate of [tex]\(\sqrt{5} + 2\)[/tex] is [tex]\(\sqrt{5} - 2\)[/tex]. So,
[tex]\[
\frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{\sqrt{5} - 2}{(\sqrt{5})^2 - (2)^2} = \frac{\sqrt{5} - 2}{5 - 4} = \frac{\sqrt{5} - 2}{1} = \sqrt{5} - 2
\][/tex]
### Step 2: Subtracting the Rationalized Expressions
Now, we subtract the simplified expressions:
[tex]\[
(\sqrt{5} + 2) - (\sqrt{5} - 2)
\][/tex]
Distribute the subtraction:
[tex]\[
\sqrt{5} + 2 - \sqrt{5} + 2 = 2 + 2 = 4
\][/tex]
So, the simplified form of the given expression is [tex]\(4\)[/tex].
### Conclusion
The correct answer is not directly provided in the options, which suggests a need for reassessment of the problem. Considering typical examination contexts, this step-by-step calculation's conclusion of [tex]\(\boxed{4}\)[/tex] may have an issue in the provided options. However, given the exact and clear arithmetic simplification, the proper mathematical conclusion stands as [tex]\(4\)[/tex].
