An infinite nested radical problem From this link, problem 36, I found that
$$\sqrt{4+\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4+\sqrt{4-...}}}}}}=2\left(\cos{\dfrac{4\pi}{19}}+\cos{\dfrac{6\pi}{19}}+\cos{\dfrac{10\pi}{19}}\right).$$
The signs : + + - + + - + + - ... .
How to prove it?
Furthermore, how to represent $\sqrt{7+2\sqrt{7-2\sqrt{7-2\sqrt{7+2\sqrt{7-2\sqrt{7-...}}}}}}$ by trigonometric function ?
The signs : + - - + - - + - - ... .
Thanks for helping.

HINT:
Let's try to show that the RHS satisfies the same equation as the LHS. 
Consider 
$$z = \cos \frac{2\pi}{19} + i \sin \frac{2\pi}{19}$$
Then $z^{19}-1 = 0$ and $z \ne 1$ so 
$$\sum_{k=0}^{18} z^k = 0$$
On the right had side you have 
$$ z^2 + z^{-2} +  z^3 + z^{-3} + z^5 + z^{-5} = z^2 + z^{17} + z^3 + z^{16} + z^5 + z^{14}$$ 
Take the polynomial equation $P(x) = 0$ obtained for the LHS and show that 
$$P(z^2 + z^{17} + z^3 + z^{16} + z^5 + z^{14})  \vdots \sum_{k=0}^{18} z^k $$
