Question

Rewrite 2tan3x in terms of tanx

Answer 1

`2\tan3x=2\tan(2x+x)=(2\tan2x+2\tan x)/(1-\tan2x\tan x)`

Use the same identity to expand
`\tan2x`.


`\tan2x=\tan(x+x)=(\tan x+\tan x)/(1-\tan x\tan x)=(2\tan x)/(1-\tan^2x)`


`\implies2\tan3x=(2(2\tan x)/(1-\tan^2x)+2\tan x)/(1-(2\tan x)/(1-\tan^2x)\tan x)`

`2\tan3x=\frac{2\tan x\left(\frac2{1-\tan^2x}+1\right)}{1-(2\tan^2x)/(1-\tan^2x)}`

`2\tan3x=(2\tan x\left(2+1-\tan^2x\right))/(1-\tan^2x-2\tan^2x)`

`2\tan3x=(2\tan x(3-\tan^2x))/(1-3\tan^2x)`