C = \(\sqrt{4+\sqrt{10+2\sqrt{5}}-\sqrt{4-\sqrt{10+2\sqrt{5}}}}\)\(C^2=4+\sqrt{10+2\sqrt{5}}+2\)\(\sqrt{4+\sqrt{10+2\sqrt{5}}-\sqrt{4-\sqrt{10+2\sqrt{5}}}}\)
+ \(4-\sqrt{10+2\sqrt{5}}\)
\(=8+2.\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\left(4-\sqrt{10+2\sqrt{5}}\right)}\)
= \(8+2\)\(\sqrt{16-10-2\sqrt{5}}\)
= \(8+2\)\(\sqrt{6-2\sqrt{5}}=8-\sqrt{5-2\sqrt{5}+1}\)
= \(8+2\sqrt{\left(\sqrt{5}-1\right)^2}\)
= \(8+2\left(\sqrt{5}-1\right)\) = \(8+2\sqrt{5}-2=6+2\sqrt{5}\)
= \(5+2\sqrt{5}+1=\left(\sqrt{5}+1\right)^2\)
C2 = \(\left(\sqrt{5}+1\right)^2\)
=> C = \(\sqrt{5}+1\)
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