\[ \text { Prove that: } \int_{0}^{\frac{\pi}{2}} \log \sin x d x=\...

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\[
\text { Prove that: } \int_{0}^{\frac{\pi}{2}} \log \sin x d x=\int_{0}^{\frac{\pi}{2}} \log \cos x d x=-\frac{\pi}{2} \log 2
\]
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