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Question Number 93689 by oustmuchiya@gmail.com last updated on 14/May/20

$$simply:\frac{{(cos2\mathrm{\Theta }-\mathit{i}sin2\mathrm{\Theta })}^7{(cos3\mathrm{\Theta }+\mathit{i}sin3\mathrm{\Theta })}^{-5}}{{(cos4\mathrm{\Theta }+\mathit{i}sin4\mathrm{\Theta })}^{12}{(cos5\mathrm{\Theta }-\mathit{i}sin5\mathrm{\Theta })}^{-6}}$$

Commented by PRITHWISH SEN 2 last updated on 14/May/20

$$\frac{{\left({\mathrm{e}}^{-\mathrm{i}2\theta }\right)}^7.{\left({\mathrm{e}}^{\mathrm{i}3\theta }\right)}^{-5}}{{\left({\mathrm{e}}^{\mathrm{i}4\theta }\right)}^{12}.{\left({\mathrm{e}}^{-\mathrm{i}5\theta }\right)}^{-6}}=\frac{{\mathrm{e}}^{-\mathrm{i}29\theta }}{{\mathrm{e}}^{\mathrm{i}78\theta }}={\mathrm{e}}^{-\mathbf{i}107\mathit{\theta }}$$$$=\mathbf{cos}\left(107\mathbf{\Theta }\right)-\mathit{i}\mathbf{sin}\left(107\mathbf{\Theta }\right)$$

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