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)