Simplify: $\;\;\;$ $\dfrac{\left(\cos 2 \theta - i \sin 2 \theta\right)^7 \left(\cos 3 \theta + i \sin 3 \theta\right)^{-5}}{\left(\cos 4 \theta + i \sin 4 \theta\right)^{12} \left(\cos 5 \theta - i \sin 5 \theta\right)^{-6}}$
$\begin{aligned} \dfrac{\left(\cos 2 \theta - i \sin 2 \theta\right)^7 \left(\cos 3 \theta + i \sin 3 \theta\right)^{-5}}{\left(\cos 4 \theta + i \sin 4 \theta\right)^{12} \left(\cos 5 \theta - i \sin 5 \theta\right)^{-6}} & = \dfrac{\left(e^{-i2 \theta}\right)^7 \cdot \left(e^{i 3 \theta}\right)^{-5}}{\left(e^{i 4 \theta}\right)^{12} \cdot \left(e^{-i 5 \theta}\right)^{-6}} \\\\ & = \dfrac{\left(e^{- i 14 \theta}\right) \cdot \left(e^{- i 15 \theta}\right)}{\left(e^{i 48 \theta}\right) \cdot \left(e^{i 30 \theta}\right)} \\\\ & = \dfrac{e^{- i 29 \theta}}{e^{i 78 \theta}} \\\\ & = e^{- i 107 \theta} \\\\ & = \cos \left(107 \theta\right) - i \sin \left(107 \theta\right) \end{aligned}$