Rational trig identities?

[Albert Chan]

Let θ = atan(1/n).
We do not know tan(nθ), but we do know tan(θ) = 1/n

Use sum of tangent formula, tan(A+B) = (tan(A) + tan(B)) / (1 - tan(A)*tan(B))

tan(2θ) = 2*tan(θ) / (1 - tan(θ)^2) = 2n/(n^2-1)
tan(3θ) = tan(2θ+θ) = (tan(2θ) + tan(θ)) / (1 - tan(2θ)*tan(θ)) = (3n^2-1)/(n^3-3*n)
...

When angle reached nθ, we have the solution.
Of course, there are more efficient ways to build tan(nθ) (see post #4)

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More food for thought, what is the sequence converging to ?
n → inf is same as x = 1/n → 0

atan(x) = (x - x^3/3 + x^5/5 - ...) = x*(1 - x^2/3 + x^4/5 - ...)

limit(atan(x)/x, x=0) = 1

For finite n: tan(nθ) < tan(1) = 1.5574077246549 ...