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

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Answer 1 · 2016-11-09T08:17:38

Please see below.

$(1/2)sqrt(2+sqrt(2+sqrt2))$

= $sqrt((2+sqrt(2+sqrt2))/4)$

= $sqrt((1+sqrt((2+sqrt2))/2)/2)$

= $sqrt((1+sqrt((2+sqrt2)/4))/2)$

= $sqrt((1+sqrt((1+sqrt(2/4))/2))/2)$

= $sqrt((1+sqrt((1+sqrt(1/2))/2))/2)$

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