Sum of Reciprocals of Squares of Odd Integers/Proof 4

Proof
Applying the substitution:
 * $\tuple {x, y} = \tuple {\dfrac {\map \sin u} {\map \cos v}, \dfrac {\map \sin v} {\map \cos u} }$

the Jacobian determinant is:

Under this substitution, the image of the region $\closedint 0 1^2$, that is the unit square, is an isosceles triangle, $\bigtriangleup$ with base/height $\dfrac \pi 2$.

(This may be observed by sketching the region $\set {\tuple {u, v}: u, v \ge 0 \land \map \cos u \ge \map \sin v \land \map \cos v \ge \map \sin u}$.)

Hence,: