Calculate $\;$ $\cos \left[\dfrac{\pi}{10} \left(\log_3 \dfrac{1}{9} + \log_{\frac{1}{9}} 3\right)\right]$
$\begin{aligned} \cos \left[\dfrac{\pi}{10} \left(\log_3 \dfrac{1}{9} + \log_{\frac{1}{9}} 3\right)\right] & = \cos \left[\dfrac{\pi}{10} \left(\log_3 3^{-2} + \dfrac{\log_3 3}{\log_3 \dfrac{1}{9}}\right)\right] \\\\ & \hspace{2cm} \left\{\text{Note: }\log_n m = \dfrac{\log_a m}{\log_a n}\right\} \\\\ & = \cos\left[\dfrac{\pi}{10} \left(\log_3 3^{-2} + \dfrac{\log_3 3}{\log_3 3^{-2}}\right)\right] \\\\ & = \cos\left[\dfrac{\pi}{10} \left(-2 \log_3 3 + \dfrac{\log_3 3}{-2 \log_3 3}\right)\right] \\\\ & \hspace{2cm} \left\{\text{Note: } \log_a m^n = n \log_a m \right\} \\\\ & = \cos \left[\dfrac{\pi}{10} \left(-2 + \dfrac{1}{-2}\right)\right] \;\; \left\{\text{Note: } \log_a a = 1 \right\} \\\\ & = \cos \left[\dfrac{\pi}{10} \times \left(\dfrac{-5}{2}\right)\right] \\\\ & = \cos \left[\dfrac{- \pi}{4}\right] \\\\ & = \cos \left[\dfrac{\pi}{4}\right] \\\\ & = \dfrac{1}{\sqrt{2}} \end{aligned}$