Finding the exact value of $\sin 30°$ using golden ratio $\sin \left(k\cdot 30^\circ\right) = \frac{\sqrt{2}}{4}\;\sqrt{\;4\;\pm_1\;\sqrt{\phi\,(a\phi+b\overline{\phi})}\;\pm_2\;\sqrt{\overline{\phi}\,(c\phi+d\overline{\phi})}\;}$
I'm currently trying to solve this for $a = 4, b = 0, c = 0, d = 4$ and $\pm_1 = -$, $\pm_2 = +$ 
where $\phi := \dfrac{\sqrt 5+ 1}{2}$ and $\overline{\phi} := \phi^{-1} = \frac{1}{2}(\sqrt{5}-1)$
There's an outer square root, which made me confused. Hence I couldn't begin to solve it.
Regards!

Note that when $b=c=0$ the nested square roots simplify down to $4\phi^2$ and $4\bar\phi^2$ so the expression simplifies to $$\frac{\sqrt{2}}{4}\sqrt{4-2\phi +2\bar\phi}$$And so the $\sqrt{5}$ terms cancel out.
