calculus - Evaluating $int_0^{pi/2} int_0^{pi/2} frac{cos(x)}{ cos(a cos(x) cos(y))} dx dy $

Can we avoid the use of the geometric interpretation combined with polar coordinates change of variable for proving that

$$\int_0^{\pi/2} \int_0^{\pi/2} \frac{\cos(x)}{ \cos(a \cos(x) \cos(y))} d x dy =\frac{\pi}{ 2a}\log\left( \frac{\displaystyle 1+ \tan \left(\frac{a}{2}\right)}{ \displaystyle 1-\tan\left(\frac{a}{2}\right)}\right)$$ ?
EDIT
What if we go further and we also consider the case

$$\int_0^{\pi/2}\int_0^{\pi/2} \int_0^{\pi/2} \frac{\cos(x)}{ \cos(a \cos(x) \cos(y) \cos(z))} d x dy \ dz$$
? What can we say about the closed form of this one?