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

We have to prove that ##(1-sin2x)/(cos2x)=(cos2x)/(1+sin2x)##
To do this we transform left side:
##(1-sin2x)/(cos2x)=((sin^2x+cos^2x)-2sinxcosx)/(cos2x)##
##=(sin^2x-2sinxcosx+cos^2x)/(cos^2x-sin^2x)##
##=(sinx-cosx)^2/((cosx-sinx)(cosx+sinx))##
##=((sinx-cosx)^2)/(-(sinx-cosx)(sinx+cosx))##
##=(cosx-sinx)/(cosx+sinx)##
Now we expand the expresion by multiplying both numerator and denominator by ##(cosx+sinx)##
So we get:
##((cosx-sinx)(cosx+sinx))/((cosx+sinx)^2)##
##=(cos^2x-sin^2x)/(cos^2x+2cosxsinx+sin^2x)##
##=(cos2x)/(1+sin2x)##