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:
\[\begin{aligned} \sqrt{2  \sqrt{3}} &= \sqrt{a}(\sqrt{b}c) \\ &= \sqrt{a (b2c\sqrt{b}+c^2)} \\ &= \sqrt{ab+ac^22ac\sqrt{b}} \end{aligned}\]Therefore,
\[\begin{aligned} ab+ac^2 &=2\\ 2ac &= 1\\ b &= 3. \end{aligned}\]Expressing \(c\) in terms of \(a\) gives the quadratic equation \(12a^28a+1=0\) 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 approach^{1} 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.

This form is described in Tutorial Algebra Volume 1 ↩