Starting with the "basic cycle" for tan(theta)tan(θ) i.e. for theta in [-pi/2,+pi/2]θ∈[−π2,+π2]
Then consider what values of xx would place (x+pi)(x+π) in this same range, i.e. (x+pi) in [-pi/2,+pi/2](x+π)∈[−π2,+π2]
(Yes; I know: because tantan has a cycle length of piπ we could recognize that the cycle will repeat at exactly the same place, but let's do this for the more general case.)
(x+pi)in[-pi/2,pi/2]color(white)("X"rarrcolor(white)("X")x in [-(3pi)/2,-pi/2](x+π)∈[−π2,π2]X→Xx∈[−3π2,−π2]
Giving us the "basic cycle" for tan(x+pi)tan(x+π):
Adding 55 to this to get y=5+tan(x+pi)y=5+tan(x+π) simply shifts the points upward 55 units:
In the image below, I have added the "non-basic cycles" as well as the "basic cycle" used for analysis:
