<s>[INST] Simplify $\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}} + \frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}$ Question: Simplify $\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}} + \frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}$
My Attempt:
On rationalizing both fractions separately and then adding (since denominator becomes $\sqrt{3}$) I got $$-2\sqrt{6}+2\sqrt{2+\sqrt{3}}+2\sqrt{2-\sqrt{3}}$$
However the given answer is $$\frac{\sqrt{6}}{3}$$
Even more confusing is that on inputting the problem into wolframalpha the solution is given as $$\sqrt{2}$$
I have broken my head over this for a couple of hours and I just can't find a solution. Hope someone can help.
 [/INST] Hint
$$2-\sqrt3=\dfrac{(\sqrt3-1)^2}2$$
$$\implies\sqrt2-\sqrt{2-\sqrt3}=\dfrac{2-(\sqrt3-1)}{\sqrt2}$$
$$\implies\dfrac{2-\sqrt3}{\sqrt2-\sqrt{2-\sqrt3}}=\sqrt{\dfrac23}\cdot\dfrac{2-\sqrt3}{\sqrt3-1}$$
$$\dfrac{2+\sqrt3}{\sqrt2+\sqrt{2+\sqrt3}}=?$$
Now take the sum
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