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Question
Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
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Solution
L.H.S = `(tan(pi/4 + x))/(tan(pi/4 - x))` now tan (A + B) = `(tan A + tanB)/(1 -tan A tan B)`
And tan (A - B) = `(tanA - tanB)/(1 + tan A tan B)`
`= ((tan pi/4 + tan x)/(1 - tan pi/4 tan x))/((tan pi/4 - tan x)/(1 + tan pi/4 tan x))`
= 1 + tan x
= `(1 - tan x)/(1 - tan x)`
= 1 + tan x
(∵ `tan pi/4 = 1` )
= `((1 + tan x) xx (1 + tan x) = (1 + tan x))^2/(1 - tan x xx 1 - tan x = ( 1- tan x)^2`
= `((1+ tan x)/(1 - tan x))^2` R.H.S
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