\( \Large log \cos x \sin x \ge 2 => \sin x \le \cos^{2}x \)

=>\( \Large sinx \le 1 - sin^{2}x \) 

=> \( \Large \sin^{2}x + \sin x - 1 \le 0 \)

=> \( \Large \left(\sin x + \frac{1}{2}\right)^{2} - \frac{5}{4} \le 0 \)

Also by definition of logarithm

\( \Large \sin x > 0, \cos x > 0, \cos x ≠ 1 \)

=> \( \Large \sin x + \frac{1}{2} \le \frac{\sqrt{5}}{2} \)

=> \( \Large 0 < \sin x \le \frac{\sqrt{5}-1}{2} \)