\( \displaystyle \lim_{x \to \frac{\pi}{2}} \tan^{2} x \left( \left(2 \sin^{2} x + 3 \sin x + 4 \right)^{\frac{1}{2}} - \left(\sin^{2} x + 6 \sin x + 2 \right)^{\frac{1}{2}} \right) \) is equal to
\( \frac{1}{12} \)
\( -\frac{1}{18} \)
\( -\frac{1}{12} \)
\( \frac{1}{6} \)
