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$$\sqrt[3]{\cos x - j \sin x}$$ is equal to
$$ (\cos x)^{\frac{1}{3}} - j(\sin x)^{\frac{1}{3}} $$
$$\sqrt[3]{\cos^2x -j\sin^2x}$$
$$ \cos \frac{x}{3} - j \sin \frac{x}{3}$$
$$ \sin \frac{x}{3} - j \cos \frac{x}{3}$$
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