Question

What are the functions `f_1, f_2`? The values ​​of double signs are defined by the function `S(k)`.

`f_1 = \sqrt {2_1 - \sqrt {2_2 \pm \sqrt {2_3 \pm \ldots \pm \sqrt {2_n}}}}`
`f_2 = \sqrt {2_1 + \sqrt {2_2 \pm \sqrt {2_3 \pm \ldots \pm \sqrt {2_n}}}}`
`S(k) = \pm 1, k = 2, 3, \ldots, n - 1`

Answer 1

The answer is detailed
here
"Equations with nested radicals ..." (pdf)

Explanation:

`f_1 = \sin (\frac {90 ^ \circ (2a + 1)} {2 ^ n})`
`f_2 = \cos (\frac {90 ^ \circ (2a + 1)} {2 ^ n})`

where

If `n = 1, 2` then `a = 0`
If `n \ge 2` then `0 \le a \le 2 ^ {n - 2} - 1`

(`n, a` are integers)

If `n, a` are known, then the signs are given by the relationships:

`r = round(a / 2 ^ {n - k})`

`S(k) = (- 1) ^ r` for `k = 2, 3, \ldots, n - 1`

Thanks!