EDITED

 

clearly note that [tanx] = 0 for all x < 3pi/12 and =1 for allx from pi/4 to tan-1(2)

 

from tan-1(2) to arctan(3) gif is 2

 

from arctan(3) to tan75 [] = 3

 

note that tan75 = 2+sqrt(3) so its gif is 3

 

ans is tan-1(2) - pi/4 + 2 tan-1(3) - 2 tan-1(2) + 3(5pi/12) - 3 tan-1(3) = pi - tan-12 - tan-13 = pi - tan-1-1 = pi/4