Rational trig identities?

[John Keith]

Surprisingly (to me), tan(n * arctan(1/n)) is rational when n is an integer. However, I can't get Mathematica nor the Prime CAS nor the 50g in exact mode to express this formula as a rational expression. Am I missing something obvious (probably!) or are such simplifications beyond the abilities of CAS's?

An iterative program is shown at A348140 but no formula is given. Here is a translation for the 49g/50g (exact mode):

Code:


\<< DUP INV \-> n s
  \<< s 1 n  1 -
    START DUP s + SWAP n / 1 SWAP - / EVAL
    NEXT
  \>>
\>>