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.