[Maxima] [Fwd: Trig puzzle]
fateman at cs.berkeley.edu
Sun Aug 2 17:52:39 CDT 2009
-------- Original Message --------
Subject: Trig puzzle
Date: Thu, 30 Jul 2009 03:33:26 +0000 (UTC)
From: rwg at sdf.lonestar.org
To: math-fun <math-fun at mailman.xmission.com>
<mailman.35692.1248585699.4407.math-fun at mailman.xmission.com>
Last night, Mathematica embarrassed Macsyma and me by simplifying
tan(3*pi/14)+cot(pi/7)-tan(pi/14) to sqrt(7) in front of dozens of
impressionable children. Later I found a tricky proof: The quotient
of two specializations of the very handy formula
(where T[n](x):=cos(n*acos(x)) = then nth Chebychev polynomial)
has the limit, with n=7,
= x^6-5*x^4+3*x^2-1/7 ,
(giving that the product of these three tans is the negative reciprocal
of their sum).
Equating coefficients of x, adjoining tan(3*pi/14)+cot(pi/7)-tan(pi/14) = y,
and eliminating the trigs gives
and the correct root can be selected numerically from the eight.
Can someone suggest a method more likely to be Mathematica's?
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