Question

How do you prove `(1 - tanx) / (1 + tanx) = (1 - sin2x) / (cos2x)`?

Answer 1

Multiply the left side in the numerator and denominator by 1-tanx and simplify to get the answer:

`(1-tanx)^2 / (1-tan^2 x)`

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

= `(sec^2x -2tanx)/((cos^2x -sin^2x)/cos^2x)` (because `1+tan^2x = sec^2x`)

=`(1-2tan x cos^2x)/cos(2x)` (because `cos^2x- sin^2x=cos 2x`)

=`(1-sin2x)/cos(2x)`