First, differentiate y. We'll call the differential y', the same as dy/dx. Multiply top and bottom of the fraction by sqrt(1+sin2x) and we get sqrt(1-sin^2(2x))/(1+sin2x) = cos2x/(1+sin2x). We can differentiate this using the formula (vdu-udv)/v^2 for differentiating a fraction u/v. So y'=(-2sin2x(1+sin2x)-2cos^2(2x))/(1+sin2x)^2. This becomes (-2sin2x-2sin^2(2x)-2cos^2(2x))/(1+sin2x)^2, giving -2(1+sin2x)/(1+sin2x)^2, simplifying to -2/(1+sin2x).
cos((pi)/4-x)=cosx/sqrt(2)+sinx/sqrt(2)=(cosx+sinx)/sqrt(2). Square this to get cos^2 and we get (1+2sinxcosx)/2=(1+sin2x)/2. The reciprocal is 2/(1+sin2x) and this is sec^2((pi)/4-x), which, added to y' above =0 (QED).