Rational trig identities?
10-10-2021, 08:02 PM
Post: #3
 Albert Chan Senior Member Posts: 2,109 Joined: Jul 2018
RE: Rational trig identities?
General case, by expanding tan(n*x), in terms of t = tan(x)

tan(n*x) = (t + tan((n-1)*x)) / (1 - t*tan((n-1)*x))

tan(2*x) = (2t) / (1 - t^2)
tan(3*x) = (3t - t^3) / (1 - 3t^2)
tan(4*x) = (4t - 4t^3) / (1 - 6t^2 + t^4)
tan(5*x) = (5t - 10t^3 + t^5) / (1 - 10t^2 + 5t^4)
...

Pattern appeared, (t+1)^n, odd powers goes on top, even powers bottom, alternate sign.

tan(n*x) = (n*t - binom(n,3)*t^3 + ...) / (1 - binom(n,2)*t^2 + ...)

Better with cotangent c = 1/t: (with x = atan(1/n), we have c = n)

tan(n*x) = (n*c^(n-1) - binom(n,3)*c^(n-3) + ...) / (c^n - binom(n,2)*c^(n-2) + ...)

Or, just let CAS do the work

CAS> a(n) := texpand(tan(n*x)) (tan(x)=1/n)
CAS> apply(a, range(1,9))

$$\displaystyle \left[ 1,\frac{4}{3},\frac{13}{9}, \frac{240}{161}, \frac{719}{475}, \frac{42372}{27755}, \frac{92567}{60319}, \frac{14970816}{9722113} \right]$$
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 Messages In This Thread Rational trig identities? - John Keith - 10-10-2021, 04:42 PM RE: Rational trig identities? - Albert Chan - 10-10-2021, 06:21 PM RE: Rational trig identities? - Albert Chan - 10-10-2021 08:02 PM RE: Rational trig identities? - Albert Chan - 10-12-2021, 04:05 PM RE: Rational trig identities? - Albert Chan - 10-10-2021, 09:25 PM RE: Rational trig identities? - John Keith - 10-11-2021, 01:08 PM RE: Rational trig identities? - Albert Chan - 10-12-2021, 02:09 PM

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