What is 1+tan2x / 1-tan2x?

1 Answer
Mar 25, 2018

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)#