if tan x=a/a+1 and tan y=1/2a+1 prove that x+y= ?/4.

tanx = a/(a+1).

tany = 1/(2a+1).

We know that tan (x+y) =  (tanx+tany)/(1-tanx*tany) is an identity.

=> tan(x+y) = {a/(a+1) +1/(2a+1)}/{ 1- a/(a+1)(2a+1)}.

=> tan(x+y) = {a(2a+1)+a+1}/{(2a+1)(a+1)-1}.

=> tan (x+y) = (2a^2+2a+1)/{2a^2+3a+1-a}.

=> tan(x+y) = (2a^2+2a+1)/(2a^2+2a+1) = 1.

=> tan(x+y) = 1.

Therefore x+y = arc tan 1 = pi/4.

Therefore x+y = pi/4