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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)##