A circle-tangent polynomial is a polynomial of degree \$N\ge3\$ or above that is tangent to the unit circle from inside at all of its N-1 intersection points. The two tails that exits the circle are considered tangent at their intersection points from inside as well. You may consider such polynomials are wrapped inside the unit circle except for the two tails.
The first several circle-tangent polynomials are shown here.
The mirror images of a circle-tangent polynomial along x- or y- axis are also circle-tangent polynomials.
Write a full program or function that, given a whole number input \$N\ge3\$, outputs a circle-tangent polynomial. You may either output the whole polynomial or only its coefficient in a reasonable order. Since such coefficients can be irrational, floating point inaccuracy within reasonable range is allowed.
The algorithm you used should be theoretically able to calculate with any valid input provided, although in reality the program could timeout.
Since this is a code-golf challenge, the shortest code of each language wins.
Both exact forms and approximate forms are shown for reference. The polynomials here are shown with positive leading terms, but mirrored counterparts around the axes are acceptable. The approximate forms are shown with 6 decimal places, but the calculation should be as accurate as possible.
Input -> Output 3 -> 8√3/9 x^3 - √3 x 1.539601 x^3 - 1.732051 x 4 -> 27/8 x^4 - 9/2 x^2 + 1 3.375 x^4 - 4.5 x^2 + 1 5 -> 2048√15/1125 x^5 - 128√15/45 x^3 + √15 x 7.050551 x^5 - 11.016486 x^3 + 3.872983 x 6 -> 3125/216 x^6 - 625/24 x^4 + 25/2 x^2 - 1 14.467593 x^6 - 26.041667 x^4 + 12.5 x^2 - 1 7 -> 1492992√35/300125 x^7 - 62208√35/6125 x^5 + 216√35/35 x^3 - √35 x 29.429937 x^7 - 60.086121 x^5 + 36.510664 x^3 - 5.916080 x
NO HARDCODING POLYNOMIALS. NO USE OF DEFAULT LOOPHOLES.