Rational trig identities?

[Albert Chan]

(10-10-2021 08:02 PM)Albert Chan Wrote:  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 + ...)

What has this property ? Complex number !
odd powers of i goes imaginery, even powers of i goes real, i*i = -1

Assuming |θ| < pi/2, we have atan(tan(θ)) = θ

z = 1 + tan(θ)*i = r * cis(θ)
Z = z^n = r^n * cis(nθ)
tan(nθ) = im(Z) / re(Z)

If θ = atan(1/n), tan(θ) = 1/n
To keep Z parts integer, we scale up Z: (1+i/n)^n → (n+i)^n

CAS> (i+2)^2       → 3+4i → a(2) = 4/3
CAS> (i+3)^3       → 18+26i → a(3) = 26/18 = 13/9
CAS> (i+4)^4       → 161+240i → a(4) = 240/161
CAS> (i+5)^5       → 1900+2876i → a(5) = 2876/1900 = 719/475