Let $f(x)= \log_e (\sin x), x ∈ (0,π)$, then which of the following statements is/are TRUE?

(A) f(x) is increasing on (0, π/2)
(B) f(x) is decreasing on (π/2, π)
(C) f(x) is increasing on (0, π)
(D) f(x) is decreasing on (0, π)

Choose the correct answer from the options given below:

(A) and (B) only

The correct answer is Option (2) → (A) and (B) only

$f(x)=\log_e(\sin x),\;x\in(0,\pi)$

$f'(x)=\frac{d}{dx}\log(\sin x)=\frac{\cos x}{\sin x}=\cot x$

$\cot x>0\text{ on }(0,\frac{\pi}{2})\Rightarrow f \text{ increasing on }(0,\frac{\pi}{2})$

$\cot x<0\text{ on }(\frac{\pi}{2},\pi)\Rightarrow f \text{ decreasing on }(\frac{\pi}{2},\pi)$

Correct Option: (A), (B)