Recently I am investigating some geometric construction (i.e., compass and ruler constructions) and encounter into the issue of finding sine for some angles. I remember most whole number degree angles could be expressed in surd forms, only it is less pretty than the standard angles of 30°, 45°, 60°. Mostly I would start from the standard angles and break down to the degree I am interested by half angle formula or angle sum and difference formula. But some angle could result in surd under surd, for example:

$\sqrt{2 - \sqrt{3}}$

Whether we can simplify this or express it in another format is the issue.

Let us start with the example. First consider if we want to factor it out into a product. Obviously it is not possible to be a product of two surd forms, so let us assume

$\sqrt{2 - \sqrt{3}} = \sqrt{a}(\sqrt{b}-c).$

We use $$\sqrt{b}-c$$ instead of $$\sqrt{b}+c$$ because we observed the negative sign in $$-\sqrt{3}$$. We may also try with $$\sqrt{2 - \sqrt{3}} = \sqrt{a}(c-\sqrt{b})$$ but we will know in the following if this is actually the case. Move the bracket term to under the surd gives:

\begin{aligned} \sqrt{2 - \sqrt{3}} &= \sqrt{a}(\sqrt{b}-c) \\ &= \sqrt{a (b-2c\sqrt{b}+c^2)} \\ &= \sqrt{ab+ac^2-2ac\sqrt{b}} \end{aligned}

Therefore,

\begin{aligned} ab+ac^2 &=2\\ 2ac &= 1\\ b &= 3. \end{aligned}

Expressing $$c$$ in terms of $$a$$ gives the quadratic equation $$12a^2-8a+1=0$$ and solving this gives $$(a,b,c)=(\frac{1}{2},3,1)$$ or $$(a,b,c)=(\frac{1}{6},3,3)$$. These two, indeed, corresponds to

$\sqrt{2 - \sqrt{3}} = \sqrt{\frac{1}{2}}(\sqrt{3}-1) = \sqrt{\frac{1}{6}}(3-\sqrt{3})$

which are the same (and explains why we assume the form $$\sqrt{a}(\sqrt{b}-c)$$ is enough).

Another approach1 is not to factor it out but express into sum of two surds:

$\sqrt{2 - \sqrt{3}} = \sqrt{a} - \sqrt{b}.$

Again, we take $$\sqrt{a}-\sqrt{b}$$ instead of $$\sqrt{a}+\sqrt{b}$$ because we observed the negative sign of $$-\sqrt{3}$$. This time, we squared both side and found that $$2 - \sqrt{3} = a + b - 2\sqrt{ab}$$. By solving $$a+b=2$$ and $$4ab=3$$, we have $$(a,b)=(\frac{1}{2},\frac{3}{2})$$ or $$(a,b)=(\frac{3}{2},\frac{1}{2})$$. Because we take only positive values for the surd, comparing the expression for correct polarity we found that the solution should be

$\sqrt{2 - \sqrt{3}} = \sqrt{\frac{3}{2}} - \sqrt{\frac{1}{2}}.$

It turns out, simplifying surds directly does not seems possible but we can easily do it if we first assume the form we want to obtain. Here contains a list of sine values for all integer angles from 0° to 90°, but the surd forms are ugly. Probably we can simplify them using this approach.

1. This form is described in Tutorial Algebra Volume 1