Question

Prove that, tan3x = 3tanx - tan³x / 1 - 3tan²x

Answer 1

We know that,

Tan 2x = 2tan x /(1 - tan² x) ---------------------------------------(1)

Add tan x on both sides.

Therefore, tan x + tan 2x = tan x + (2tan x /(1 - tan² x))

= (tan x - tan³ x + 2tan x)/(1 - tan² x)

tan x + tan 2x = (3tan x - tan³ x) / (1 - tan² x) ------------------------(2)

tan 3x can also be written as tan (x + 2x)

We know the formula , tan (a+b) = (tan a + tan b) / (1 - tan a * tan b)

So, tan (x + 2x) = (tan x + tan 2x) / (1 - tan x * tan 2x)

Replace the values from equation (1) and (2)

tan 3x = (3tan x - tan³ x)/(1 - tan² x) × 1/((1 - 2tan² x)/(1 - tan² x))

= (3tan x - tan³ x)/(1 - tan² x * (1 - tan² x)/(1 - tan² x - 2tan² x)

tan 3x = (3tan x - tan³ x) / (1 - 3tan² x)

Hence the proof.