Explanation

Let a=(2+sqrt(3))^x, b=(2-sqrt(3))^x. Then ab=[(2+sqrt(3))(2-sqrt(3))]^x=1^x=1. Given a+b=2 and ab=1, (a-b)^2=(a+b)^2-4ab=0, so a-b=0. The key insight is that the two bases are reciprocals.