In[540]:= $Version Out[540]= "11.0.1 for Mac OS X x86 (64-bit) (September 21, 2016)" cc[x_] := Nest[((# + √(#^2 - 4))/2)^√2 + ((# - √(#^2 - 4))/2)^√2 &, x, 2] In[653]:=FullSimplify@cc@-Sqrt[2 (1 - Sin[√2 π])] == 2 Sin[√2 π] Out[653]= 2^(-1 - Sqrt[ 2]) (((-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] - I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^Sqrt[ 2] + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] + I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^Sqrt[ 2] - \[Sqrt](-2^( 1 + Sqrt[2]) + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] - I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^( 2 Sqrt[2]) + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] + I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^(2 Sqrt[2])))^Sqrt[ 2] + ((-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] - I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^Sqrt[ 2] + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] + I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^Sqrt[ 2] + \[Sqrt](-2^( 1 + Sqrt[2]) + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] - I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^( 2 Sqrt[2]) + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] + I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^(2 Sqrt[2])))^Sqrt[2]) == 2 Sin[Sqrt[2] \[Pi]] In[654]:= Chop@N@# & /@ % Out[654]= True --Bill Gosper