A functional identity involving elliptic integrals

We show that the following double integral ∫ _0^πd x∫ _0^xd y1/√(1- [b]pcos x)√(1+ [b]qcos y) remains invariant as one trades the parameters p and q for p'=√(1-p^2) and q'=√(1-q^2) , respectively. This invariance property is suggested from symmetry considerations in the operating characteristics of a semiconductor Hall effect device.