If $z = \cos heta + i\sin heta$, find an expression for $z^n + \dfra…

Question

If $z = \cos	heta + i\sin	heta$, find an expression for $z^n + \dfrac{1}{z^n}$.

Accepted Answer

C. $2\cos n	heta$

**Apply De Moivre.** $z^n = \cos n	heta + i\sin n	heta$.
**Compute the reciprocal.** Since $|z| = 1$, $\dfrac{1}{z^n} = \overline{z^n} = \cos n	heta - i\sin n	heta$ (conjugate of a unit-modulus complex number is its inverse).
**Sum.** $z^n + \dfrac{1}{z^n} = 2\cos n	heta$ — the imaginary parts cancel.
**Symmetric companion.** Subtracting instead gives $z^n - \dfrac{1}{z^n} = 2i\sin n	heta$ (option B) — useful for deriving multiple-angle formulae, but it's not what the question asks for.

Answer

$\sin 2n	heta$

Answer

$\cos n	heta$

Answer

$2i\sin n	heta$

If $z = \cos\theta + i\sin\theta$, find an expression for $z^n + \dfrac{1}{z^n}$.