Find the tangent of the sum of inverse tangents
Python 2, 76 72 bytes

from fractions import*
f=lambda k:Fraction(k and(k+f(k-1))/(1-k*f(k-1)))


Use the identity:

tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A) * tan(B))


We have:

f(k) = 0                                    if k = 0
     = (k + f(k - 1)) / (1 - k * f(k - 1))  if k &gt; 0


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Thanks to Luis Mendo, save 4 bytes.