c ( 1 + 1 ) = c c − s s s ( 1 + 1 ) = c s + c s = 2 c s c ( 1 + 1 + 1 ) = c ( 1 + 1 ) c − s ( 1 + 1 ) s = c c c − c s s − 2 c s s = c c c − 3 c s s s ( 1 + 1 + 1 ) = s ( 1 + 1 ) c + c ( 1 + 1 ) s = 2 c c s + c c s − s s s = 3 c c s − s s s 0 = s ( 1 + 1 + 1 + 1 + 1 ) = s ( 1 + 1 + 1 ) c ( 1 + 1 ) + c ( 1 + 1 + 1 ) s ( 1 + 1 ) 0 = ( 3 c c s − s s s ) ( c c − s s ) + ( c c c − 3 c s s ) 2 c s 0 = ( 3 c c − s s ) ( c c − s s ) + ( c c c − 3 c s s ) 2 c 0 = ( 3 c c c c − 3 c c s s − c c s s + s s s s ) + ( 2 c c c c − 6 c c s s ) 0 = 5 c c c c − 10 c c s s + s s s s 0 = 5 c c − 10 c s + s s s = 1 − c 0 = 5 c c − 10 c + 10 c c + 1 − 2 c + c c 0 = 16 c c − 12 c + 1 0 = c c − 3 4 c + 1 16 c = 3 8 + 9 64 − 4 64 = 3 8 + 1 8 5 c = 6 16 + 1 16 20 = 1 4 6 + 20 {\displaystyle {\begin{matrix}c(1+1)=cc-ss\\s(1+1)=cs+cs=2cs\\c(1+1+1)=c(1+1)c-s(1+1)s=ccc-css-2css=ccc-3css\\s(1+1+1)=s(1+1)c+c(1+1)s=2ccs+ccs-sss=3ccs-sss\\0=s(1+1+1+1+1)=s(1+1+1)c(1+1)+c(1+1+1)s(1+1)\\0=(3ccs-sss)(cc-ss)+(ccc-3css)2cs\\0=(3cc-ss)(cc-ss)+(ccc-3css)2c\\0=(3cccc-3ccss-ccss+ssss)+(2cccc-6ccss)\\0=5cccc-10ccss+ssss\\0=5cc-10cs+ss\\s=1-c\\0=5cc-10c+10cc+1-2c+cc\\0=16cc-12c+1\\0=cc-{\frac {3}{4}}c+{\frac {1}{16}}\\c={\frac {3}{8}}+{\sqrt {{\frac {9}{64}}-{\frac {4}{64}}}}={\frac {3}{8}}+{\frac {1}{8}}{\sqrt {5}}\\{\sqrt {c}}={\sqrt {{\frac {6}{16}}+{\frac {1}{16}}{\sqrt {20}}}}={\frac {1}{4}}{\sqrt {6+{\sqrt {20}}}}\end{matrix}}}
z.z.
6 + 20 = 1 + 5 6 + 20 = 1 + 2 ∗ 5 + 5 20 = 2 ∗ 5 {\displaystyle {\begin{matrix}{\sqrt {6+{\sqrt {20}}}}=1+{\sqrt {5}}\\6+{\sqrt {20}}=1+2*{\sqrt {5}}+5\\{\sqrt {20}}=2*{\sqrt {5}}\end{matrix}}}
q.e.d.
