On Wed, Dec 2, 2020 at 4:18 PM Bill Gosper <billgosper@gmail.com> wrote:
{Cos[π/17]==1/16 (1-√17+√(34-2 √17)+2 √(17+3 √17+√(170+38 √17))),
Sin[π/34]==1/16 (-1+√17+√(34-2 √17)-2 √(17+3 √17-√(170+38 √17)))}
or even
2 √(17 + 3 √17 - √(170 + 38 √17)) == √( 2 (3 + √17) (2 √17 - √(2 (17 - √17))))
et al. I.e., {Cos[π/17]==1/16 (1-√17+√(34-2 √17)+2 √(17+3 √17+√(170+38 √17)))==1/16 (1-√17+√(34-2 √17)+√(2(3+√17)(2√17+√(2(17-√17))))), Sin[π/34]==1/16 (-1+√17+√(34-2 √17)-2 √(17+3 √17-√(170+38 √17)))==1/16 (-1+√17+√(34-2 √17)-√( 2 (3 + √17) (2 √17 - √(2 (17 - √17)))))} You might suspect that FullSimplify is sufficiently exhaustive, and I am sufficiently nefarious, to concoct these with a jiggered SimplifyCount function which favors "17"s. I confess to trying, but that didn't seem to do anything. —rwg
https://mathworld.wolfram.com/TrigonometryAnglesPi17.html includes a messier version of Cos[π/17].
(People keep forgetting that Sin[π/5] is messier than Sin[π/10].) —rwg