Sum of Reciprocals of Squares of Odd Integers/Proof 4

Proof
Applying the substitution
 * $\displaystyle \left( {x, y} \right) = \left( \frac{\sin\left( {u} \right)} {\cos\left( {v} \right)}, \frac{\sin\left( {v} \right)} {\cos\left( {u} \right)}\right)$

The Jacobian determinant is,

Under this substitution, the image of the region $\left[ {0, 1} \right]^2$, ie. the unit square, is an isosceles triangle, $\bigtriangleup$ with base/height $\frac \pi 2$. (this may be observed by sketching the region $\{ \left( {u, v} \right): u, v \ge 0 \land \cos \left( {u} \right) \ge \sin \left( {v} \right) \land \cos \left( {v} \right) \ge \sin \left( {u} \right) \}$) Hence,