How to simplify this irrational fraction?

Question

I was solving a geometry question when I arrived at this-

$$\theta=\cot^{-1}\frac {\sqrt2-\sqrt {2-\sqrt3}}{\sqrt {2-\sqrt3}}$$

Now,the answer given is $30^0$.So,the above fraction must simplify to $\sqrt3$.

But I cannot simplify it.How to do it?

Answer 1

Hint: multiply by $\sqrt{2+\sqrt{3}}$

Hint: $\sqrt{4+2\sqrt{3}} = \sqrt{(1+\sqrt{3})^2}$

Answer 2

Let us check your claim that

$$\frac {\sqrt2-\sqrt {2-\sqrt3}}{\sqrt {2-\sqrt3}}=\sqrt3.$$ We can simplify as $$\frac{\sqrt2}{\sqrt{2-\sqrt3}}=\sqrt3+1.$$

and by squaring,

$$\frac2{2-\sqrt3}=3+2\sqrt3+1=2(2+\sqrt3).$$

You can easily conclude.