Prove the following: tan (π/4 + x) / tan (π/4 - x) = [(1 + tan x) / (1 - tan x)]²

Solution:

LHS = tan (π/4 + x) / tan (π/4 - x)

= [(tan π/4 + tan x) / (1 - tan π/4 tan x)] / [(tan π/4 - tan x) / (1 + tan π/4 tan x)]   

 [because tan (A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B) ]

= [(1 + tan x) / (1 - tan x)] / [(1 - tan x)/(1 + tan x)]   

 [(by trigonometric table, tan π/4 = 1]

= [(1 + tan x) / (1 - tan x)] × [(1 + tan x)/(1 - tan x)]

= [(1 + tan x) / (1 - tan x)]²

= RHS

NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.3 Question 7

Prove the following: tan (π/4 + x) / tan (π/4 - x) = [(1 + tan x) / (1 - tan x)]²

Summary:

We got, tan (π/4 + x) / tan (π/4 - x) = [(1 + tan x) / (1 - tan x)]². Hence Proved