Question

Prove that tan4x=4tanx(1-tan^2x)/1-6tan^2x+tan^4x

Answer 1

Tan 4x = Tan 2(2x) = 2 Tan 2x / (1 - Tan² 2x) = 2{2 tan x /(1-tan²x)}/{1-4 Tan²x/(1-tan²x)²}
At this step, we can cancel one tan²x
= 4 tan x(1 - tan²x)/{(tan^4 x - 2 tan²x + 1) - 4tan²x}
=4 tan x(1- tan²x)/(1 - 6 tan²x + tan^4 x)

Answer 2

L.H.S.

tan 4x = tan 2(2x)

[We know that tan 2x = 2 tan x / 1 - tan 2 x]

= 2 tan 2x / 1 - tan2 (2x)

[now putting tan 2x = 2 tan x / 1 - tan2x]

= 2[2 tan x/1-tan2x] / 1 - [2 tan x / 1 - tan2x] 2

=[4 tan x / 1 - tan 2 x] / [1 - 4 tan 2 x / (1 - tan2 x)2]

=[4 tan x / 1 - tan 2 x] / [ (1- tan 2 x)2 - 4 tan 2 x / (1 - tan2 x)2]

= 4 tan x (1 - tan 2 x) / (1- tan 2 x)2 - 4 tan 2 x

= 4 tan x (1 - tan 2 x) / 1 - 2 tan2 x +tan 4 x - 4tan2 x

= [4 tan x (1 - tan 2 x) / 1 - 6 tan 2 x + tan 4 x].