A question by Ramanujan about a relational expression of a triangle

I found the following question in a book without any proof:

Question : Suppose that each length of three edges of a triangle $ABC$ are $BC=a, CA=b, AB=c$ respectively. If $$\frac1a=\frac1b+\frac1c, \frac2a=\frac{1}{c-b}-\frac{1}{c+b},$$ then prove $$\sqrt[3]{\cos{2A}}+\sqrt[3]{\cos{2B}}+\sqrt[3]{\cos{2C}}=\sqrt[3]{\frac{5-3\color{blue}{\sqrt[3]{7}}}{2}}\ .$$

This book says that this is the question by Ramanujan.

I've tried to prove this, but I'm facing difficulty. Can anyone help? If you have any helpful information, please let me know it.

Solution 1:

Given,

$$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$

then,

$$F_p = x_1^{1/3}+x_2^{1/3}+x_3^{1/3} = \sqrt[3]{-(6n+1)+3\sqrt[3]{np}}\tag2$$

where $p = 9n^2+3n+1$. This is a special case of an identity by Ramanujan. Let $n=-1$, change variables $x \to t$, and we have Lucian's,

$$t^3+t^2-2t-1=0$$

Hence,

$$t_1^{1/3}+t_2^{1/3}+t_3^{1/3} = \sqrt[3]{5-3\sqrt[3]{7}}$$

We deduce,

$$t_1 = 2\cos 2A,\quad t_2 = 2\cos 2B,\quad t_3 = 2\cos 2C$$

so,

$$\sqrt[3]{\cos{2A}}+\sqrt[3]{\cos{2B}}+\sqrt[3]{\cos{2C}}=\sqrt[3]{\frac{5-3\color{blue}{\sqrt[3]{7}}}{2}}$$

correcting a typo in the original question. (It should be $\color{blue}{\sqrt[3]{7}}$.)