A) \( \Large \left[ 0, \frac{1}{2} \right] \) |
B) \( \Large \left[\frac{\sqrt{5}-1}{2}, 1 \right] \) |
C) \( \Large \left[ 0, \frac{\sqrt{5}-1}{2} \right] \) |
D) none of these |
C) \( \Large \left[ 0, \frac{\sqrt{5}-1}{2} \right] \) |
\( \Large log \cos x \sin x \ge 2 => \sin x \le \cos^{2}x \)
=>\( \Large sinx \le 1 - sin^{2}x \)
=> \( \Large \sin^{2}x + \sin x - 1 \le 0 \)
=> \( \Large \left(\sin x + \frac{1}{2}\right)^{2} - \frac{5}{4} \le 0 \)
Also by definition of logarithm
\( \Large \sin x > 0, \cos x > 0, \cos x ≠ 1 \)
=> \( \Large \sin x + \frac{1}{2} \le \frac{\sqrt{5}}{2} \)
=> \( \Large 0 < \sin x \le \frac{\sqrt{5}-1}{2} \)