$$\begin{gathered} \mathrm{I} = {\int }_0^{\frac{\mathrm{\pi }}{2}} \left( 2 \log \sin \mathrm{x} - \log \sin 2\mathrm{x} \right) \mathrm{dx} \\ \mathrm{I} = {\int }_0^{\frac{\mathrm{\pi }}{2}} \left( \log \frac{{\sin }^2 \mathrm{x} }{2 \sin \mathrm{x} . \cos \mathrm{x}} . \mathrm{dx} \right) \\ \mathrm{I} = {\int }_0^{\frac{\mathrm{\pi }}{2}} \log \left( \frac{\tan \mathrm{x} }{2}\right) . \mathrm{dx} .............\left(\mathrm{i}\right) \\ \mathrm{Using} \mathrm{property} {\int }_0^{\mathrm{a}}\mathrm{f} ( \mathrm{x} ) \mathrm{dx} = {\int }_0^{\mathrm{a}}\mathrm{f} ( \mathrm{a} - \mathrm{x} ) \mathrm{dx} \end{gathered}$$

We get,

$$\begin{gathered} \mathrm{I} = {\int }_0^{\frac{\mathrm{\pi }}{2}} \log \left( \frac{\tan \left({\displaystyle \frac{\mathrm{\pi }}{2}} - \mathrm{x} \right)}{2}\right) \mathrm{dx} \\ \Rightarrow \mathrm{I} = {\int }_0^{\frac{\mathrm{\pi }}{2}} \log \left(\frac{\cot \mathrm{x}}{2}\right) \mathrm{dx} ........\left(\mathrm{ii}\right) \end{gathered}$$

Adding (i) and (ii)

$$\begin{gathered} 2\mathrm{I} = {\int }_0^{\frac{\mathrm{\pi }}{2}} \left[ \log \left(\frac{\tan \mathrm{x}}{2}\right) + \log \left(\frac{\cot \mathrm{x}}{2}\right) \right] \mathrm{dx} \\ \Rightarrow 2\mathrm{I} = {\int }_0^{\frac{\mathrm{\pi }}{2}} \log \left[\left(\frac{\tan \mathrm{x}}{2}\right) \left(\frac{\cot \mathrm{x}}{2}\right) \right] \mathrm{dx} \\ \Rightarrow \mathrm{I} = \frac{1}{2}{\int }_0^{\frac{\mathrm{\pi }}{2}} \log \left(\frac{1}{4}\right) \mathrm{dx} \\ \Rightarrow \mathrm{I} = \frac{1}{2} \log \left(\frac{1}{4}\right) \mathrm{x} \left(\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow \mathrm{I} = \frac{1}{2} \log {\left(\frac{1}{4}\right)}^2 \mathrm{x} \left(\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow \mathrm{I} = \log \left(\frac{1}{2}\right) \mathrm{x} \left(\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow \mathrm{I} = \frac{\mathrm{\pi }}{2} \log \frac{1}{2} \end{gathered}$$