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\frac{2}{\sqrt{3}+1}
Divide 1 by \frac{\sqrt{3}+1}{2} by multiplying 1 by the reciprocal of \frac{\sqrt{3}+1}{2}.
\frac{2\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}
Rationalize the denominator of \frac{2}{\sqrt{3}+1} by multiplying numerator and denominator by \sqrt{3}-1.
\frac{2\left(\sqrt{3}-1\right)}{\left(\sqrt{3}\right)^{2}-1^{2}}
Consider \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{2\left(\sqrt{3}-1\right)}{3-1}
Square \sqrt{3}. Square 1.
\frac{2\left(\sqrt{3}-1\right)}{2}
Subtract 1 from 3 to get 2.
\sqrt{3}-1
Cancel out 2 and 2.