0

1 + i

$$-$$ 1 + i

1 $$-$$ i

$${z^n} + {1 \over {{z^n}}} = 2\cos \left( {{{n\pi } \over 2} - n\theta } \right)$$

$${z^n} + {1 \over {{z^n}}} = 2\sin \left( {{{n\pi } \over 2} - n\theta } \right)$$

$${z^n} - {1 \over {{z^n}}} = 2i\sin \left( {n\theta - {{n\pi } \over 2}} \right)$$

$${z^n} - {1 \over {{z^n}}} = 2i\cos \left( {{{n\pi } \over 2} - n\theta } \right)$$