The value of ${\left(\cos 2\mathrm{\theta }+\mathrm{i} \sin 2\mathrm{\theta }\right)}^5{\left(\sin \mathrm{\theta }+\mathrm{i} \mathrm{cos\theta }\right)}^4$is

$\begin{array}{rcl} & & {\left(\cos 2\mathrm{\theta }+\mathrm{i} \sin 2\mathrm{\theta }\right)}^5 {\left(\sin \mathrm{\theta }+\mathrm{i} \mathrm{cos\theta }\right)}^4\\ & & (\cos 10\mathrm{\theta }+\mathrm{i} \sin 10\mathrm{\theta }) {\left(-{\mathrm{i}}^2\mathrm{sin\theta }+\mathrm{i} \mathrm{cos\theta }\right)}^4\\ & & \left(\mathrm{cis}10\mathrm{\theta }\right) {\mathrm{i}}^4 {(\mathrm{cos\theta }-\mathrm{i} \mathrm{sin\theta })}^4= \left(\mathrm{cis}10\mathrm{\theta }\right) \mathrm{cis}(-4\mathrm{\theta })\\ & =& \mathrm{cis}10\mathrm{\theta } \mathrm{cis}(-4\mathrm{\theta })\\ & =& \mathrm{cis} (10\mathrm{\theta }-4\mathrm{\theta })=\mathrm{cis}6\mathrm{\theta }\\ & =& \cos 6\mathrm{\theta }+\mathrm{i} \sin 6\mathrm{\theta }\end{array}$