If tan x = a/(a+1) and tan y = 1/(2a+1) prove that x+y=pi/4.

x + y = pi/4 if and only if tan (x+y) = tan pi/4 =
1

So, we'll have to prove that tan (x+y) =
1.

We'll apply the formula of tangent of the sum of 2
angles:

tan (x + y) = (tan x + tan y)/(1 - tan x*tan y)
(1)

We know that tan x = a/(a+1) and tan y = 1/(2a+1) and
we'll substitute them in (1).

tan (x + y) = [a/(a+1) +
1/(2a+1)]/{1 -[a/(a+1)]*[1/(2a+1)]}

tan (x + y) =
[(2a^2+a+a+1)/(a+1)(2a+1)]/{[(a+1)(2a+1) -
a]/(a+1)(2a+1)]}

We'll combine like terms ad we'll
simplify:

tan (x + y) = (2a^2+2a+1)/(2a^2 + 3a + 1 -
a)

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

We notice that the numerator and denominator are
equal:

tan (x + y) = 1
q.e.d.

Since tan (x + y) = 1, then x + y =
pi/4.