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$$(\cos 5 \theta - i \sin 5 \theta)^{2}$$ is same as
$$\cos 10 \theta + 1 \sin 10 \theta$$
$$\cos 25 \theta - i \sin 25 \theta$$
$$(\cos \theta + 1 \sin \theta)^{-10}$$
$$(\cos \theta - i \sin \theta )^{-10}$$
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