Question

If x+y=π/4 ,prove that(1+tanx)(1+tany)=2

Answer 1

Solution:

Given:
or, x + y = pi/4
So:
or, tan(x+y) = tan(pi/4)
or, tan(x+y) = 1

We know that:
[tan(pi/4) = 1]
[tan(x+y) = (tanx+tany)/(1-tanx.tany)]

Now:
or, (tanx+tan y)/(1 - tan x.tany) = 1

By Cross multiplying we get:

or, tanx+tany = 1-tanx.tany

or, tanx + tany + tanx.tany = 1

[By Adding '1' on both sides we have]:

or, 1 + tanx + tany + tanx.tany = 1 + 1

or, 1(1+tanx) + tany(1+tanx) = 2

or, (1 + tan x) ( 1 + tan y ) = 2
L.H.S = R.H.S.
(Proved)
(Answer)