We can attempt to write the angle as a sum or difference of angles. We use angles we can easily find tangent of.
tan(4π/12 + 3π/12) =
tan(π/3 + π/4)
Now we can use the addition angle identity for tangent:
tan(x + y) = (tanx + tany) / (1 - tanxtany)
tan(π/3 + π/4) = [tan(π/3) + tan(π/4)] / [1 - tan(π/3)tan(π/4)]
Use the fact that
tan(π/3) = sin(π/3) / cos(π/3)
= (√(3) / 2) / (1 / 2)
tan(π/4) = sin(π/4) / cos(π/4)
= (√(2) / 2) / (√(2) / 2)
Finish up here.