Explanation

With p=cos^2(theta)/sin(theta), q=sin^2(theta)/cos(theta): p^2q^2=sin^2(theta)cos^2(theta) and p^2+q^2=(cos^6(theta)+sin^6(theta))/(sin^2(theta)cos^2(theta))=(1-3sin^2(theta)cos^2(theta))/(sin^2(theta)cos^2(theta)). Thus p^2+q^2+3=1/(sin^2(theta)cos^2(theta)), and p^2q^2(p^2+q^2+3)=1. This is a standard identity; the stored key is confirmed.