More puzzles for FullSimplify: Sin[π/22] == 1/10 + √11 (Cos[1/5 ArcTan[5/Sqrt[877 + 1958/√5]]] - Cos[π/5 - 1/5 ArcTan[5/109 Sqrt[4385 + 1958 √5]]])/5 == 1/10 + 2/5 √11 Sin[1/10 (π- ArcTan[5/199 Sqrt[71045 - 26378 √5]])] Sin[ 1/10 (π - ArcTan[5/199 Sqrt[71045 + 26378√5]])] "Explanation": Sin[π/22] satisfies a (solvable) quintic. But isn't π/22 preferable to ArcTan[...]/5? Not if you want the radicals. Whatever happened to Mathematica's quintic solver? I have something that seems to solve (the solvable) sextics and septics as well. —rwg