Proof $\frac{-\sqrt{2 - \sqrt{2 - \sqrt{2}}}}{\sqrt2} + \frac{\sqrt{2 + \sqrt2}}{\sqrt2 \sqrt{2 - \sqrt{2 - \sqrt2}}} =\sqrt{2-\sqrt{2+\sqrt2}}$

Question

I want to prove the following equation

$$\frac{-\sqrt{2 - \sqrt{2 - \sqrt{2}}}}{\sqrt{2}} + \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2} \sqrt{2 - \sqrt{2 - \sqrt{2}}}}$$ I know this must be equal to $\sqrt{2-\sqrt{2+\sqrt{2}}}$, but I find it pretty hard to prove it manually, without requiring a calculator nor estimates (since the expression should be equal to one of the roots of the polinomyal $f(x)=x^8-8x^6+20x^4-16x^2+2$, and all of them are real and distinct).

Thanks in advance for your help.

Answer 1

Note that the second term of the LHS can be expressed as

$$\begin{align} \frac{\sqrt{2 + \sqrt{2} }}{\sqrt{2} \sqrt{2 - \sqrt{2 - \sqrt{2}}}} =\frac{\sqrt{2 + \sqrt{2 - \sqrt{2}}}} {\sqrt{2}} \end{align}$$ which can be verified by cross multiply. Then, $$LHS = \sqrt{ \frac{2 + \sqrt{2 - \sqrt{2}}} {2} } - \sqrt{ \frac{2 - \sqrt{2 - \sqrt{2}}} {2} }=\sqrt{2-\sqrt{2+\sqrt{2}}}=RHS $$

where the denest formula below is used

$$\sqrt{a-\sqrt c} =\sqrt{\frac{a+\sqrt {a^2-c}}{2}} -\sqrt{\frac{a-\sqrt {a^2-c}}{2}} $$

with $a=2$ and $c=2+\sqrt2$.

Answer 2

$a=\sqrt{2-\sqrt{2-\sqrt{2}}}+\sqrt{2-\sqrt{2+\sqrt{2}}}$

$a^2=4+\sqrt{2}$

$a=\sqrt{2(2+\sqrt{2})}$

$b=\sqrt{2-\sqrt{2-\sqrt{2}}}*\sqrt{2-\sqrt{2+\sqrt{2}}}=\sqrt{4+\sqrt{2}-2(\sqrt{2-\sqrt{2-\sqrt{2}}}+\sqrt{2-\sqrt{2+\sqrt{2}}})}=\sqrt{4+\sqrt{2}-2(\sqrt{2(2+\sqrt{2})})} $

$-\sqrt{2-\sqrt{2-\sqrt{2}}}/\sqrt{2}+\sqrt{2+\sqrt{2}}/\left(\sqrt{2}\sqrt{2-\sqrt{2-\sqrt{2}}}\right)=\sqrt{2-\sqrt{2+\sqrt{2}}}?$ $-\sqrt{2-\sqrt{2-\sqrt{2}}}+\sqrt{2+\sqrt{2}}/\left(\sqrt{2-\sqrt{2-\sqrt{2}}}\right)=\sqrt{2(2-\sqrt{2+\sqrt{2}}})$

$\sqrt{2+\sqrt{2}}-(2-\sqrt{2-\sqrt{2}})=\sqrt{2(2-\sqrt{2+\sqrt{2}})(2-\sqrt{2-\sqrt{2}})}$

$\sqrt{2-\sqrt{2-\sqrt{2}}}+\sqrt{2-\sqrt{2+\sqrt{2}}}-2=\sqrt{4+\sqrt{2}-2(\sqrt{2(2+\sqrt{2})})}$

$\sqrt{2(2+\sqrt{2})}-2=\sqrt{2(4+\sqrt{2}-2(\sqrt{2(2+\sqrt{2})})}$

$\sqrt{2+\sqrt{2}}-\sqrt{2}=\sqrt{4+\sqrt{2}-2(\sqrt{2(2+\sqrt{2})})}$

Squared:

$4+\sqrt{2}-2(\sqrt{2(2+\sqrt{2})})=4+\sqrt{2}-2(\sqrt{2(2+\sqrt{2})})$

The proof is done.