Other values of n for which the minimal nesting of radicals is 2 or less include n=40 and n=120. For instance for n=40, we have:
sin(π40)=(1−√58)√2+√2+√28√2−√2√5+√5
cos(π40)=(√5−18)√2−√2+√28√2+√2√5+√5
sin(3π40)=(−√5−18)√2−√2+√28√2+√2√5−√5
cos(3π40)=(√5+18)√2+√2+√28√2−√2√5−√5
sin(7π40)=(√5+18)√2+√2−√28√2−√2√5−√5
cos(7π40)=(√5+18)√2−√2+√28√2+√2√5−√5
sin(9π40)=(√5−18)√2+√2+√28√2−√2√5+√5
cos(9π40)=(1−√58)√2−√2+√28√2+√2√5+√5
sin(11π40)=(1−√58)√2−√2+√28√2+√2√5+√5
cos(11π40)=(√5−18)√2+√2+√28√2−√2√5+√5
cos(11π40)=(√5−18)√2+√2+√28√2−√2√5+√5
sin(13π40)=(1+√58)√2−√2+√28√2+√2√5−√5
cos(13π40)=(√5+18)√2+√2−√28√2−√2√5−√5
sin(17π40)=(1+√58)√2+√2+√28√2+√2√5−√5
cos(17π40)=(−√5−18)√2−√2+√28√2+√2√5−√5
sin(19π40)=(√5−18)√2−√2+√28√2+√2√5+√5
cos(19π40)=(1−√58)√2+√2+√28√2−√2√5+√5
cos(13π40)=(√5+18)√2+√2−√28√2−√2√5−√5
sin(17π40)=(1+√58)√2+√2+√28√2+√2√5−√5
cos(17π40)=(−√5−18)√2−√2+√28√2+√2√5−√5
sin(19π40)=(√5−18)√2−√2+√28√2+√2√5+√5
cos(19π40)=(1−√58)√2+√2+√28√2−√2√5+√5
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