Question: cos(Pi/17) in terms of exact radicals

How can I find the exact solution? > a20:=cos(Pi/17) = sqrt(sqrt(38*sqrt(17) + 170) + 3*sqrt(17) + 17)/8 + sqrt(34 - 2*sqrt(17))/16 - sqrt(17)/16 + 1/16: > evalf(a20); 0.9829730997 = 0.9829730994 I found cos(Pi/5) and cos(Pi/15). a1:=cos(Pi - 2*Pi/5) = cos(3*Pi/5): > a2:=cos(Pi - 2*x) = cos(3*x); a2 := -cos(2 x) = cos(3 x) > a3:=expand(a2); 2 3 a3 := -2 cos(x) + 1 = 4 cos(x) - 3 cos(x) > a4:=subs(cos(x)=y,a3); 2 3 a4 := -2 y + 1 = 4 y - 3 y > a5:=[solve(a4,y)]; 1/2 1/2 5 5 a5 := [-1, ---- + 1/4, 1/4 - ----] 4 4 > a6:=cos(Pi/5)=a5[2]; a6 := cos(1/5*Pi) = 1/4*5^(1/2)+1/4 > evalf(a6); 0.8090169943 = 0.8090169942 > a7:=cos(Pi/3-2*Pi/5)=cos(Pi/15): > a8:=cos(Pi/3-2*x)=cos(Pi/15): > a9:=expand(a8); a9 := cos(x)^2-1/2+3^(1/2)*sin(x)*cos(x) = cos(1/15*Pi) > a10:=subs(sin(x)=sqrt(1-cos(x)^2),a9); a10 := cos(x)^2-1/2+3^(1/2)*(1-cos(x)^2)^(1/2)*cos(x) = cos(1/15*Pi) > a11:=subs(cos(x)=a5[2],a10): > a12:=simplify(expand(a11)); a12 := -1/8+1/8*5^(1/2)+1/16*3^(1/2)*(10-2*5^(1/2))^(1/2)*5^(1/2)+1/16*3^(1/2)*(10-2*5^(1/2))^(1/2) = cos(1/15*Pi) > evalf(a12); 0.9781476005 = 0.9781476007 I have a simplier version. > a13:=sqrt(6*sqrt(5) + 30)/8 + sqrt(5)/8 - 1/8 = cos(Pi/15): > evalf(a13); 0.9781476010 = 0.9781476007