Methods to simplify $\int _0^{\frac{\pi }{2}}\int _0^{\frac{\pi }{2}}\frac{\ln (\cos (x))\ln (\cos (y))}{\sin (x)+\sin (y)}\:dx\:dy.$

Question

I want to evaluate: $$\int _0^{\frac{\pi }{2}}\int _0^{\frac{\pi }{2}}\frac{\ln \left(\cos \left(x\right)\right)\ln \left(\cos \left(y\right)\right)}{\sin \left(x\right)+\sin \left(y\right)}\:dx\:dy.$$ But I've no idea on how to simplify this into $1$ integral. I tried to write as follows: $$\int _0^{\frac{\pi }{2}}\ln \left(\cos \left(y\right)\right)\int _0^{\frac{\pi }{2}}\csc \left(x\right)\ln \left(\cos \left(x\right)\right)\:dx\:dy$$ $$-\int _0^{\frac{\pi }{2}}\ln \left(\cos \left(y\right)\right)\sin \left(y\right)\int _0^{\frac{\pi }{2}}\frac{\ln \left(\cos \left(x\right)\right)}{\sin \left(x\right)\left(\sin \left(x\right)+\sin \left(y\right)\right)}\:dx\:dy.$$ But I still have trouble with the $2$nd double integral.