We know that ,
#color(red)((1)arc tanx+arc tany=arc tan((x+y)/(1-x*y))# , #color(red)(x*y < 1#
Let , #A=2arc tan(1/2)+arc tan (1/7)#
Using #(1)# we get ,
#color(blue)(2arc tan(1/2))=arc tan(1/2)+arc tan(1/2)#
#color(white)(2arc tan(1/2))=arc tan((1/2+1/2)/(1-1/2*1/2)),## x*y=1/2*1/2#=#1/4<1#
#color(white)(2arc tan(1/2))=arc tan(1/(1-1/4))#
#color(white)(2arc tan(1/2))=arc tan(1/(3/4))#
#color(blue)(2arc tan(1/2)=arc tan(4/3)#
So,
#A=color(blue)(arc tan (4/3))+arc tan(1/7)tocolor(red)( Apply(1)#
#A=arc tan((4/3+1/7)/(1-4/3*1/7)) , x*y=4/3*1/7=4/21<1#
#A=arc tan((28+3)/(21-4))#
#A=arc tan(31/17)#
