Question

What is 1+tan2x / 1-tan2x?

Answer 1

There are sevral answers like:
`color(red)((1+tan2x)/(1-tan2x)=tan(pi/4+2x)........to(M)`
`color(red)((1+tan2x)/(1-tan2x)=(tan^2x-2tanx-1)/(tan^2x+2tanx-1)........to(N)`,
pls mention what to prove.

Explanation:

We know that, `color(red)(tan(pi/4)=1, and`

`color(blue)(tan(A+B)=(tanA+tanB)/(1-tanAtanB)....to(1)`
So,
`(1+tan2x)/(1-tan2x)=(tan(pi/4)+tan2x)/(1-tan(pi/4)tan2x)`
Using (1), we get

`color(red)((1+tan2x)/(1-tan2x)=tan(pi/4+2x)........to(M)`

Again ,if we use ,

`tan2x=(2tanx)/(1-tan^2x)`,then

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

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

`color(red)((1+tan2x)/(1-tan2x)=(tan^2x-2tanx-1)/(tan^2x+2tanx-1)........to(N)`