Most of us know that $\displaystyle \int^{\pi/2}_{0} \ln (\sin x) \ dx = \int^{\pi/2}_{0} \ln (\cos x) \ dx = - \frac{\pi}{2} \ln 2$

But what about $\displaystyle \int^{\pi/2}_{0} \ln (\sin x) \ln (\cos x) \ dx $?

The worked solution I've seen is quite confusing.