From one of the maths seminar course introduction to calculus for the first year students @King math's school of academic.
$$\int_{0}^{\pi/2}\int_{0}^{\pi/2}\frac{\mathrm dx \mathrm dy}{\left(\frac{\tan x}{\sin y}\right)^2+\left(\frac{\sin x}{\cos y}\right)^2}=-\frac{\pi}{15}$$
I try several days to simplify this but, couldn't.
Can anyone show how this is done to get to the result proposed?