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)##