Integral with specific method
We can use the same method used in this answer. $$ \begin{align} &\int_0^{\pi/2}\log(\sin(x))\log(\cos(x))\,\mathrm{d}x\\ &=\int_0^{\pi/2}\left(\log(2)+\sum_{j=1}^\infty\frac{\cos(2jx)}{j}\right)\left(\log(2)+\sum_{k=1}^\infty(-1)^k\frac{\cos(2kx)}{k}\right)\,\mathrm{d}x\\ &=\frac\pi2\log(2)^2+2\log(2)\int_0^{\pi/2}\sum_{k=1}^\infty\frac{\cos(4kx)}{2k}\,\mathrm{d}x+\int_0^{\pi/2}\sum_{k=1}^\infty(-1)^k\frac{\cos^2(2kx)}{k^2}\,\mathrm{d}x\\ &=\frac\pi2\log(2)^2+\frac\pi4\sum_{k=1}^\infty\frac{(-1)^k}{k^2}\\ &=\frac\pi2\log(2)^2-\frac{\pi^3}{48} \end{align} $$