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

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)##