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

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### 2 Answers

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