cos formulas
The following cos formulas can be found (starting with the most trivial ones):
`cos(pi/2) = sqrt(0)/2 = 0`
`cos(pi/3) = sqrt(1)/2`
`cos(pi/4) = sqrt(2)/2`
`cos(pi/6) = sqrt(3)/2`
`cos(0) = sqrt(4)/2 = 1`
Now let's go to a little more complicated ones:
`cos(1*pi/8) = sqrt(2+sqrt(2))/2`
`cos(2*pi/8) = sqrt(2)/2`
`cos(3*pi/8) = sqrt(2-sqrt(2))/2`
Nicely symmetric, as you can see. Next ones are:
`cos(1*pi/12) = sqrt(2+sqrt(3))/2`
`cos(2*pi/12) = sqrt(3)/2`
`cos(3*pi/12) = sqrt(2)/2`
`cos(4*pi/12) = sqrt(1)/2`
`cos(5*pi/12) = sqrt(2-sqrt(3))/2`
Nice. Symmetric again. Next ones up:
`cos(1*pi/16) = sqrt(2+sqrt(2+sqrt(2)))/2`
`cos(2*pi/16) = sqrt(2+sqrt(2))/2`
`cos(3*pi/16) = sqrt(2+sqrt(2-sqrt(2)))/2`
`cos(4*pi/16) = sqrt(2)/2`
`cos(5*pi/16) = sqrt(2-sqrt(2-sqrt(2)))/2`
`cos(6*pi/16) = sqrt(2-sqrt(2))/2`
`cos(7*pi/16) = sqrt(2-sqrt(2+sqrt(2)))/2`
Seems like we can go on like this. Next set:
`cos(1*pi/24) = sqrt(2+sqrt(2+sqrt(3)))/2`
`cos(2*pi/24) = sqrt(2+sqrt(3))/2`
`cos(3*pi/24) = sqrt(2+sqrt(2))/2`
`cos(4*pi/24) = sqrt(3)/2`
`cos(5*pi/24) = sqrt(2+sqrt(2-sqrt(3)))/2`
`cos(6*pi/24) = sqrt(2)/2`
`cos(7*pi/24) = sqrt(2-sqrt(2-sqrt(3)))/2`
`cos(8*pi/24) = sqrt(1)/2`
`cos(9*pi/24) = sqrt(2-sqrt(2))/2`
`cos(10*pi/24) = sqrt(2-sqrt(3))/2`
`cos(11*pi/24) = sqrt(2-sqrt(2+sqrt(3)))/2`
This strategy seems to work as long we double the denominator, like so:
`cos(pi/48) = sqrt(2+sqrt(2+sqrt(2+sqrt(3))))/2`
`cos(pi/96) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))/2`
`cos(pi/192) = sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))/2`
Now, let's look with which number series we can do this. We can definitely start with 1 and double up. The numbers will then be `2^n`:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
We can also start with 3. The numbers will then be:
3, 6, 12, 24, 48, 96, 192, 384, 768, ...
It is also possible to start with 5. The resulting series will then be:
5, 10, 20, 40, 80, 160, 320, 640, ...
And when starting with 15, the number series will be:
15, 30, 60, 120, 240, 480, 960, ...
The starting numbers seem to have to give odd-sided constructible polygons
Thus, our choices are 1,3,5,15,17,51,85 and 255.
For 17, the number series we get is:
17, 34, 68, 136, 272, 544, ...
For 51, it yields the series:
51, 102, 204, 408, 816, ...
For 85 it gives us the series:
85, 170, 340, 680, 1360, ...
We finish up with the series for 255:
255, 510, 1020, 2040, ...
Now, let's look for the formulas for these (the numerator multiplier for `pi` has to be always smaller than half of the denominator inside cos):
Let's start with 1 for all the multipliers in front of `pi` and the first series
`cos(pi/4) = 1/2 sqrt(2)`
`cos(pi/8) = 1/2 sqrt(2+sqrt(2))`
`cos(pi/16) = 1/2 sqrt(2+sqrt(2+sqrt(2)))`
`cos(pi/32) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2))))`
`cos(pi/64) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))`
`cos(pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))`
`cos(pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(pi/512) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))))`
If we include different multipliers for `pi`, the first series results in;
`cos(1*pi/4) = 1/2 sqrt(2)`
`cos(1*pi/8) = 1/2 sqrt(2+sqrt(2))`
`cos(3*pi/8) = 1/2 sqrt(2-sqrt(2))`
`cos(1*pi/16) = 1/2 sqrt(2+sqrt(2+sqrt(2)))`
`cos(3*pi/16) = 1/2 sqrt(2+sqrt(2-sqrt(2)))`
`cos(5*pi/16) = 1/2 sqrt(2-sqrt(2-sqrt(2)))`
`cos(7*pi/16) = 1/2 sqrt(2-sqrt(2+sqrt(2)))`
`cos(1*pi/32) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2))))`
`cos(3*pi/32) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2))))`
`cos(5*pi/32) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2))))`
`cos(7*pi/32) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2))))`
`cos(9*pi/32) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2))))`
`cos(11*pi/32) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2))))`
`cos(13*pi/32) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2))))`
`cos(15*pi/32) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2))))`
`cos(1*pi/64) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))`
`cos(3*pi/64) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))`
`cos(5*pi/64) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))`
`cos(7*pi/64) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))`
`cos(9*pi/64) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))`
`cos(11*pi/64) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))`
`cos(13*pi/64) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))`
`cos(15*pi/64) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))`
`cos(17*pi/64) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))`
`cos(19*pi/64) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))`
`cos(21*pi/64) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))`
`cos(23*pi/64) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))`
`cos(25*pi/64) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))`
`cos(27*pi/64) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))`
`cos(29*pi/64) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))`
`cos(31*pi/64) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))`
`cos(1*pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))`
`cos(3*pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2))))))`
`cos(5*pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2))))))`
`cos(7*pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2))))))`
`cos(9*pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2))))))`
`cos(11*pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2))))))`
`cos(13*pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2))))))`
`cos(15*pi/128) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2))))))`
`cos(17*pi/128) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2))))))`
`cos(19*pi/128) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2))))))`
`cos(21*pi/128) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2))))))`
`cos(23*pi/128) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2))))))`
`cos(25*pi/128) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2))))))`
`cos(27*pi/128) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2))))))`
`cos(29*pi/128) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2))))))`
`cos(31*pi/128) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))`
`cos(33*pi/128) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))`
`cos(35*pi/128) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2))))))`
`cos(37*pi/128) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2))))))`
`cos(39*pi/128) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2))))))`
`cos(41*pi/128) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2))))))`
`cos(43*pi/128) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2))))))`
`cos(45*pi/128) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2))))))`
`cos(47*pi/128) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2))))))`
`cos(49*pi/128) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2))))))`
`cos(51*pi/128) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2))))))`
`cos(53*pi/128) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2))))))`
`cos(55*pi/128) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2))))))`
`cos(57*pi/128) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2))))))`
`cos(59*pi/128) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2))))))`
`cos(61*pi/128) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2))))))`
`cos(63*pi/128) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))`
`cos(1*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(3*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(5*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(7*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(9*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(11*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(13*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(15*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(17*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(19*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(21*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(23*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(25*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(27*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(29*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(31*pi/256) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(33*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(35*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(37*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(39*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(41*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(43*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(45*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(47*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(49*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(51*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(53*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(55*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(57*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(59*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(61*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(63*pi/256) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(65*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(67*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(69*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(71*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(73*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(75*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(77*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(79*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(81*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(83*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(85*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(87*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(89*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(91*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(93*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(95*pi/256) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(97*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(99*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(101*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(103*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(105*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(107*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(109*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(111*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(113*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2)))))))`
`cos(115*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(117*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(119*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(121*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2)))))))`
`cos(123*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2)))))))`
`cos(125*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2)))))))`
`cos(127*pi/256) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))`
The second series then gives us:
`cos(pi/3) = 1/2`
`cos(pi/6) = 1/2 sqrt(3)`
`cos(pi/12) = 1/2 sqrt(2+sqrt(3))`
`cos(pi/24) = 1/2 sqrt(2+sqrt(2+sqrt(3)))`
`cos(pi/48) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(3))))`
`cos(pi/96) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))`
`cos(pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))`
`cos(pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(pi/768) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))))`
If we include different multipliers for `pi`, the second series gives:
`cos(1*pi/3) = 1/2`
`cos(1*pi/6) = 1/2 sqrt(3)`
`cos(1*pi/12) = 1/2 sqrt(2+sqrt(3))`
`cos(5*pi/12) = 1/2 sqrt(2-sqrt(3))`
`cos(pi/24) = 1/2 sqrt(2+sqrt(2+sqrt(3)))`
`cos(5*pi/24) = 1/2 sqrt(2+sqrt(2-sqrt(3)))`
`cos(7*pi/24) = 1/2 sqrt(2-sqrt(2-sqrt(3)))`
`cos(11*pi/24) = 1/2 sqrt(2-sqrt(2+sqrt(3)))`
`cos(1*pi/48) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(3))))`
`cos(5*pi/48) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(3))))`
`cos(7*pi/48) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(3))))`
`cos(11*pi/48) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(3))))`
`cos(13*pi/48) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(3))))`
`cos(17*pi/48) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(3))))`
`cos(19*pi/48) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(3))))`
`cos(23*pi/48) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(3))))`
`cos(1*pi/96) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))`
`cos(5*pi/96) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))`
`cos(7*pi/96) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))`
`cos(11*pi/96) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))`
`cos(13*pi/96) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))`
`cos(17*pi/96) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))`
`cos(19*pi/96) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))`
`cos(23*pi/96) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))`
`cos(25*pi/96) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))`
`cos(29*pi/96) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))`
`cos(31*pi/96) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))`
`cos(35*pi/96) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))`
`cos(37*pi/96) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))`
`cos(41*pi/96) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))`
`cos(43*pi/96) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))`
`cos(47*pi/96) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))`
`cos(1*pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))`
`cos(5*pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3))))))`
`cos(7*pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3))))))`
`cos(11*pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3))))))`
`cos(13*pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3))))))`
`cos(17*pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3))))))`
`cos(19*pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3))))))`
`cos(23*pi/192) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3))))))`
`cos(25*pi/192) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3))))))`
`cos(29*pi/192) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3))))))`
`cos(31*pi/192) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3))))))`
`cos(35*pi/192) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3))))))`
`cos(37*pi/192) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3))))))`
`cos(41*pi/192) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3))))))`
`cos(43*pi/192) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3))))))`
`cos(47*pi/192) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))`
`cos(49*pi/192) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))`
`cos(53*pi/192) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3))))))`
`cos(55*pi/192) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3))))))`
`cos(59*pi/192) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3))))))`
`cos(61*pi/192) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3))))))`
`cos(65*pi/192) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3))))))`
`cos(67*pi/192) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3))))))`
`cos(71*pi/192) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3))))))`
`cos(73*pi/192) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3))))))`
`cos(77*pi/192) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3))))))`
`cos(79*pi/192) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3))))))`
`cos(83*pi/192) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3))))))`
`cos(85*pi/192) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3))))))`
`cos(89*pi/192) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3))))))`
`cos(91*pi/192) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3))))))`
`cos(95*pi/192) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))`
`cos(1*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(5*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(7*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(11*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(13*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(17*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(19*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(23*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(25*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(29*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(31*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(35*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(37*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(41*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(43*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(47*pi/384) = 1/2 sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(49*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(53*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(55*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(59*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(61*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(65*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(67*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(71*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(73*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(77*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(79*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(83*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(85*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(89*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(91*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(95*pi/384) = 1/2 sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(97*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(101*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(103*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(107*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(109*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(113*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(115*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(119*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(121*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(125*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(127*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(131*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(133*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(137*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(139*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(143*pi/384) = 1/2 sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(145*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(149*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(151*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(155*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(157*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(161*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(163*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(167*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(169*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(173*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(175*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(179*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(181*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2+sqrt(3)))))))`
`cos(185*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(2-sqrt(3)))))))`
`cos(187*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2-sqrt(3)))))))`
`cos(191*pi/384) = 1/2 sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3)))))))`
`cos(pi/768) = 1/2 sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(3))))))))`
The third series results in:
`cos(pi/5) = 1/2 ((1 + sqrt(5))/2)`
`cos(pi/10) = 1/2 sqrt((5 + sqrt(5))/2)`
`cos(pi/20) = 1/2 sqrt(2 + sqrt((5 + sqrt(5))/2))`
`cos(pi/40) = 1/2 sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2)))`
`cos(pi/80) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))`
`cos(pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(pi/640) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2)))))))`
If we include different multipliers for `pi`, the third series gives:
`cos(1*pi/5) = 1/2 ((1 + sqrt(5))/2)`
`cos(1*pi/10) = 1/2 sqrt((5 + sqrt(5))/2)`
`cos(3*pi/10) = 1/2 sqrt((5 - sqrt(5))/2)`
`cos(1*pi/20) = 1/2 sqrt(2 + sqrt((5 + sqrt(5))/2))`
`cos(3*pi/20) = 1/2 sqrt(2 + sqrt((5 - sqrt(5))/2))`
`cos(7*pi/20) = 1/2 sqrt(2 - sqrt((5 - sqrt(5))/2))`
`cos(9*pi/20) = 1/2 sqrt(2 - sqrt((5 + sqrt(5))/2))`
`cos(1*pi/40) = 1/2 sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2)))`
`cos(3*pi/40) = 1/2 sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2)))`
`cos(7*pi/40) = 1/2 sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2)))`
`cos(9*pi/40) = 1/2 sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2)))`
`cos(11*pi/40) = 1/2 sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2)))`
`cos(13*pi/40) = 1/2 sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2)))`
`cos(17*pi/40) = 1/2 sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2)))`
`cos(19*pi/40) = 1/2 sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2)))`
`cos(1*pi/80) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))`
`cos(3*pi/80) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))`
`cos(7*pi/80) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))`
`cos(9*pi/80) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))`
`cos(11*pi/80) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))`
`cos(13*pi/80) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))`
`cos(17*pi/80) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))`
`cos(19*pi/80) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))`
`cos(21*pi/80) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))`
`cos(23*pi/80) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))`
`cos(27*pi/80) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))`
`cos(29*pi/80) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))`
`cos(31*pi/80) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))`
`cos(33*pi/80) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))`
`cos(37*pi/80) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))`
`cos(39*pi/80) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))`
`cos(1*pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(3*pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2)))))`
`cos(7*pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2)))))`
`cos(9*pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2)))))`
`cos(11*pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2)))))`
`cos(13*pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2)))))`
`cos(17*pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2)))))`
`cos(19*pi/160) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(21*pi/160) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(23*pi/160) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2)))))`
`cos(27*pi/160) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2)))))`
`cos(29*pi/160) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2)))))`
`cos(31*pi/160) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2)))))`
`cos(33*pi/160) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2)))))`
`cos(37*pi/160) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2)))))`
`cos(39*pi/160) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(41*pi/160) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(43*pi/160) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2)))))`
`cos(47*pi/160) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2)))))`
`cos(49*pi/160) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2)))))`
`cos(51*pi/160) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2)))))`
`cos(53*pi/160) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2)))))`
`cos(57*pi/160) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2)))))`
`cos(59*pi/160) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(61*pi/160) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(63*pi/160) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2)))))`
`cos(67*pi/160) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2)))))`
`cos(69*pi/160) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2)))))`
`cos(71*pi/160) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2)))))`
`cos(73*pi/160) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2)))))`
`cos(77*pi/160) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2)))))`
`cos(79*pi/160) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2)))))`
`cos(1*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(3*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(7*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(9*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(11*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(13*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(17*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(19*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(21*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(23*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(27*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(29*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(31*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(33*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(37*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(39*pi/320) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(41*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(43*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(47*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(49*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(51*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(53*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(57*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(59*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(61*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(63*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(67*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(69*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(71*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(73*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(77*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(79*pi/320) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(81*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(83*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(87*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(89*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(91*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(93*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(97*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(99*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(101*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(103*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(107*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(109*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(111*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(113*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(117*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(119*pi/320) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(121*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(123*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(127*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(129*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(131*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(133*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(137*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(139*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(141*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
`cos(143*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(147*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(149*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(151*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 + sqrt(5))/2))))))`
`cos(153*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - sqrt((5 - sqrt(5))/2))))))`
`cos(157*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 - sqrt(5))/2))))))`
`cos(159*pi/320) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt((5 + sqrt(5))/2))))))`
The forth series gives us:
`cos(pi/15) = 1/8 (-1 + sqrt(5) + sqrt(6 (5 + sqrt(5))))`
`cos(pi/30) = 1/4 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))`
`cos(pi/60) = 1/2 sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))`
`cos(pi/120) = 1/2 sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))`
`cos(pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(pi/960) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))))`
Including different multipliers for `pi`, the forth series results in:
`cos(1*pi/15) = 1/8 (-1 + sqrt(5) + sqrt(6 (5 + sqrt(5))))`
`cos(7*pi/15) = 1/8 (-1 - sqrt(5) + sqrt(6 (5 - sqrt(5))))`
`cos(1*pi/30) = 1/4 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))`
`cos(7*pi/30) = 1/4 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))`
`cos(11*pi/30) = 1/4 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))`
`cos(13*pi/30) = 1/4 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))`
`cos(1*pi/60) = 1/2 sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))`
`cos(7*pi/60) = 1/2 sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))`
`cos(11*pi/60) = 1/2 sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))`
`cos(13*pi/60) = 1/2 sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))`
`cos(17*pi/60) = 1/2 sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))`
`cos(19*pi/60) = 1/2 sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))`
`cos(23*pi/60) = 1/2 sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))`
`cos(29*pi/60) = 1/2 sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))`
`cos(1*pi/120) = 1/2 sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))`
`cos(7*pi/120) = 1/2 sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))`
`cos(11*pi/120) = 1/2 sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))`
`cos(13*pi/120) = 1/2 sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))`
`cos(17*pi/120) = 1/2 sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))`
`cos(19*pi/120) = 1/2 sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))`
`cos(23*pi/120) = 1/2 sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))`
`cos(29*pi/120) = 1/2 sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))`
`cos(31*pi/120) = 1/2 sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))`
`cos(37*pi/120) = 1/2 sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))`
`cos(41*pi/120) = 1/2 sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))`
`cos(43*pi/120) = 1/2 sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))`
`cos(47*pi/120) = 1/2 sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))`
`cos(49*pi/120) = 1/2 sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))`
`cos(53*pi/120) = 1/2 sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))`
`cos(59*pi/120) = 1/2 sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))`
`cos(1*pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(7*pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))))`
`cos(11*pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))))`
`cos(13*pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))))`
`cos(17*pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))))`
`cos(19*pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))))`
`cos(23*pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))))`
`cos(29*pi/240) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(31*pi/240) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(37*pi/240) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))))`
`cos(41*pi/240) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))))`
`cos(43*pi/240) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))))`
`cos(47*pi/240) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))))`
`cos(49*pi/240) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))))`
`cos(53*pi/240) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))))`
`cos(59*pi/240) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(61*pi/240) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(67*pi/240) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))))`
`cos(71*pi/240) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))))`
`cos(73*pi/240) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))))`
`cos(77*pi/240) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))))`
`cos(79*pi/240) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))))`
`cos(83*pi/240) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))))`
`cos(89*pi/240) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(91*pi/240) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(97*pi/240) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))))`
`cos(101*pi/240) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))))`
`cos(103*pi/240) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))))`
`cos(107*pi/240) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5)))))))`
`cos(109*pi/240) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5)))))))`
`cos(113*pi/240) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5)))))))`
`cos(119*pi/240) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5)))))))`
`cos(1*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(7*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(11*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(13*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(17*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(19*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(23*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(29*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(31*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(37*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(41*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(43*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(47*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(49*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(53*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(59*pi/480) = 1/2 sqrt(2 + sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(61*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(67*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(71*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(73*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(77*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(79*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(83*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(89*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(91*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(97*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(101*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(103*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(107*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(109*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(113*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(119*pi/480) = 1/2 sqrt(2 + sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(121*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(127*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(131*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(133*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(137*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(139*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(143*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(149*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(151*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(157*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(161*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(163*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(167*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(169*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(173*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(179*pi/480) = 1/2 sqrt(2 - sqrt(2 - sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(181*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(187*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(191*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(193*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(197*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(199*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(203*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(209*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 - sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(211*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
`cos(217*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(221*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(223*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 - 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(227*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) - sqrt(6 (5 - sqrt(5))))))))`
`cos(229*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) - sqrt(6 (5 + sqrt(5))))))))`
`cos(233*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 - sqrt(5) + sqrt(6 (5 - sqrt(5))))))))`
`cos(239*pi/480) = 1/2 sqrt(2 - sqrt(2 + sqrt(2 + sqrt(2 + 1/2 sqrt(7 + sqrt(5) + sqrt(6 (5 + sqrt(5))))))))`
The fifth series results in:
`cos(pi/17) = sqrt((15 + sqrt(17) + sqrt(2 (17 - sqrt(17))) + sqrt(2 (34 + 6 sqrt(17) - sqrt(2 (17 - sqrt(17))) + sqrt(34 (17 - sqrt(17))) - 8 sqrt(2 (17 + sqrt(17))))))/32)`
`cos(pi/34) = sqrt((17 - sqrt(17) + sqrt(2 (17 - sqrt(17))) + sqrt(2 (34 + 6 sqrt(17) + sqrt(2 (17 - sqrt(17))) - sqrt(34 (17 - sqrt(17))) + 8 sqrt(2 (17 + sqrt(17))))))/32)`
`cos(pi/51) = 1/16 ((sqrt(5)-1)/2) (sqrt(6 (17 + sqrt(17) + sqrt(2 (17 + sqrt(17))) - sqrt(2 (34 - 6 sqrt(17) - 8 sqrt(2 (17 - sqrt(17))) + sqrt(34 (17 + sqrt(17))) + sqrt(2 (17 + sqrt(17)))))))) + `
`1/16 (sqrt(2 (15 - sqrt(17) - sqrt(2 (17 + sqrt(17))) + sqrt(2 (34 - 6 sqrt(17) - 8 sqrt(2 (17 - sqrt(17))) + sqrt(34 (17 + sqrt(17))) + sqrt(2 (17 + sqrt(17))))))))`
`cos(pi/68) = 1/2 sqrt(2+ sqrt((17 - sqrt(17) + sqrt(2 (17 - sqrt(17))) + sqrt(2 (34 + 6 sqrt(17) + sqrt(2 (17 - sqrt(17))) - sqrt(34 (17 - sqrt(17))) + 8 sqrt(2 (17 + sqrt(17))))))/8))`
`cos(pi/85) = 1/16 ((sqrt(5)-1)/2) (sqrt(15 - sqrt(17) - sqrt(2 (17 + sqrt(17))) - sqrt(2 (34 - 6 sqrt(17) - 8 sqrt(2 (17 - sqrt(17))) + sqrt(34 (17 + sqrt(17))) + sqrt(2 (17 + sqrt(17))))))) + `
`1/16 (sqrt(5 + sqrt(5))) (sqrt(17 + sqrt(17) + sqrt(2 (17 + sqrt(17))) + sqrt(2 (34 - 6 sqrt(17) - 8 sqrt(2 (17 - sqrt(17))) + sqrt(34 (17 + sqrt(17))) + sqrt(2 (17 + sqrt(17)))))))`
`cos(pi/102) = 1/8 sqrt(3/2 (15 - sqrt(17) + sqrt(2 (17 + sqrt(17))) + sqrt(2 (34 - 6 sqrt(17) + 8 sqrt(2 (17 - sqrt(17))) - sqrt(2 (17 + sqrt(17))) - sqrt(34 (17 + sqrt(17))))))) + `
`1/8 sqrt(8 - sqrt(2 (15 - sqrt(17) - sqrt(2 (17 + sqrt(17))) + sqrt(2 (34 - 6 sqrt(17) - 8 sqrt(2 (17 - sqrt(17))) + sqrt(2 (17 + sqrt(17))) + sqrt(34 (17 + sqrt(17))))))))`
`cos(pi/136) = 1/8 sqrt(1/2 (2 + sqrt(2)) (15 + sqrt(17) - sqrt(2 (17 - sqrt(17))) + sqrt(2 (34 + 6 sqrt(17) + sqrt(2 (17 - sqrt(17))) - sqrt(34 (17 - sqrt(17))) + 8 sqrt(2 (17 + sqrt(17))))))) + `
`1/8 sqrt((2 - sqrt(2)) (8 - sqrt(2 (15 + sqrt(17) + sqrt(2 (17 - sqrt(17))) - sqrt(2 (34 + 6 sqrt(17) - sqrt(2 (17 - sqrt(17))) + sqrt(34 (17 - sqrt(17))) - 8 sqrt(2 (17 + sqrt(17)))))))))`
