$$\cos n\theta +i\sin n\theta = e^{in\theta}=(e^{i\theta})^n=(\cos\theta+i\sin \theta)^n$$ $$\cos n\theta =\mathrm{Re}\,(\cos\theta+i\sin \theta)^n$$ $$\sin n\theta =\mathrm{Im}\,(\cos\theta+i\sin \theta)^n$$

$$\dots$$

References

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