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

Use the following facts:
##tanx=sinx/cosx##
##sin2x=2sinxcosx##
##cos2x=cos^2x-sin^2x##
##sin^2x+cos^2x=1##
So you get:
##(1-sinx/cosx)/(1+sinx/cosx)=(1-2sinxcosx)/(cos^2x-sin^2x)##
##(cosx-sinx)/cancel(cosx)*cancel((cosx))/cancel((cosx+sinx))=(1-2sinxcosx)/((cosx-sinx)cancel((cosx+sinx)))##
Taking: ##(cosx-sinx)## to the left side and using the fact that: ##sin^2x+cos^2x=1##
##(cosx-sinx)^2=sin^2x+cos^2x-2sinxcosx## which are indeed equals.