How to show $\sqrt{4+2\sqrt{3}}-\sqrt{3} = 1$
Hint: $$4+2\sqrt{3} = 1+2\sqrt{3}+\sqrt{3}^2 = (1+\sqrt{3})^2.$$
Here’s an approach that doesn’t require you to spot a not-terribly-obvious factorization. Multiplying by the conjugate to get rid of some of the square roots is a fairly natural thing to do, and
$$\left(\sqrt{4+2\sqrt3}-\sqrt3\right)\left(\sqrt{4+2\sqrt3}+\sqrt3\right)=1+2\sqrt3\;.$$
Thus, the desired result holds if and only if
$$1+2\sqrt3=\sqrt{4+2\sqrt3}+\sqrt3\;,$$
or
$$1+\sqrt3=\sqrt{4+2\sqrt3}\;,$$
which is easily verified.