Calculate $\;$ $\log_{\frac{1}{2}} \left[\log_3 \cos \left(\dfrac{\pi}{6}\right) - \log_3 \sin \left(\dfrac{\pi}{6}\right)\right]$
$\begin{aligned} \log_{\frac{1}{2}} \left[\log_3 \cos \left(\dfrac{\pi}{6}\right) - \log_3 \sin \left(\dfrac{\pi}{6}\right)\right] & = \log_{\frac{1}{2}} \left[\log_3 \left(\dfrac{\cos \left(\dfrac{\pi}{6}\right)}{\sin \left(\dfrac{\pi}{6}\right)}\right)\right] \\\\ & \hspace{2cm} \left\{\text{Note: } \log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y \right\} \\\\ & = \log_{\frac{1}{2}} \left[\log_3 \cot \left(\dfrac{\pi}{6}\right)\right] \\\\ & = \log_{\frac{1}{2}} \left[\log_3 \sqrt{3}\right] \\\\ & = \log_{\frac{1}{2}} \left[\log_3 3^{\frac{1}{2}}\right] \\\\ & = \log_{\frac{1}{2}} \left[\dfrac{1}{2} \log_3 3\right] \\\\ & = \log_{\frac{1}{2}} \dfrac{1}{2} \;\; \left\{\text{Note: } \log_a a = 1 \right\} \\\\ & = 1 \end{aligned}$