# How do you prove that (1/2)sqrt(2+sqrt(2+sqrt 2))=sqrt((1+sqrt((1+1/sqrt 2)/2)) /2)?

Nov 9, 2016

#### Explanation:

$\left(\frac{1}{2}\right) \sqrt{2 + \sqrt{2 + \sqrt{2}}}$

= $\sqrt{\frac{2 + \sqrt{2 + \sqrt{2}}}{4}}$

= $\sqrt{\frac{1 + \frac{\sqrt{\left(2 + \sqrt{2}\right)}}{2}}{2}}$

= $\sqrt{\frac{1 + \sqrt{\frac{2 + \sqrt{2}}{4}}}{2}}$

= $\sqrt{\frac{1 + \sqrt{\frac{1 + \sqrt{\frac{2}{4}}}{2}}}{2}}$

= $\sqrt{\frac{1 + \sqrt{\frac{1 + \sqrt{\frac{1}{2}}}{2}}}{2}}$