QPaperGen

We are given:
\[
f(x) = \log \sin x
\]

Differentiate with respect to \( x \):
\[
f'(x) = \frac{d}{dx} \left( \log \sin x \right) = \frac{1}{\sin x} \cdot \cos x = \cot x
\]

Now analyze the sign of \( f'(x) = \cot x \) in the given intervals:

\[\] In \( \left(0, \dfrac{\pi}{2} \right) \),
\( \cot x > 0 \), since both \( \sin x > 0 \) and \( \cos x > 0 \),
\( \Rightarrow f'(x) > 0 \)
\( \Rightarrow f \) is strictly increasing on \( \left(0, \dfrac{\pi}{2} \right) \).

\[\] In \( \left( \dfrac{\pi}{2}, \pi \right) \),
\( \cot x < 0 \), since \( \sin x > 0 \) and \( \cos x < 0 \),
\( \Rightarrow f'(x) < 0 \)
\( \Rightarrow f \) is strictly decreasing on \( \left( \dfrac{\pi}{2}, \pi \right) \).
\[\]
\(\textbf{Conclusion:}\)
\[
f(x) = \log \sin x \text{ is strictly increasing on } \left(0, \dfrac{\pi}{2} \right) \text{ and strictly decreasing on } \left( \dfrac{\pi}{2}, \pi \right).
\]