User:Wsjradha/Cotangent Sum Problem
Let be the twenty (complex) roots of the equation
Calculate the value of
For the purpose of this solution will be the sum of the roots of the 20th degree polynomial, taken at a time. For example,
Also, will be the sum of the cotangent inverses of the roots, taken at a time. The cotangent inverses will be multiplied as necessary, then added.
Also, will be the sum of the tangents of the cotangent inverses of the roots, taken at a time. Basically, this is the same as except that the tangents are taken right after the cotangent inverses. For example,
Let . This equals . There is a formula that states the following, where, for the purposes of this formula only, , is the sum of through , taken at a time, in the fashion described above:
When applied to this problem, it yields:
Taking the reciprocal of either side, one gets:
Multiple the numerator and the denominator of the right hand side by .
can be determined, from the original 20th degree equation using Vieta's Formulas, to be Therefore,
This simplifies to