Prove the conjugate-power identity for positive integers n.

01 / Standard#Your turn

\omega^n+\omega^{-n}=2\cos\frac{2n\pi}7,\quad\omega=\operatorname{cis}(2\pi/7)

Official paper · jm02-2022 · 4(b)(ii) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use |ω|=1 to interpret the negative power.

Hint 2

The two imaginary parts cancel.

Worked solution

The identity holds for every positive integer n.

Checks and common pitfalls: The negative power is a reciprocal; it is a conjugate because the modulus is one.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Apply De Moivre to both signs.
$\omega^{\pm n}=\cos(2n\pi/7)\pm i\sin(2n\pi/7)$
Add the conjugate expressions.
$\omega^n+\omega^{-n}=2\cos(2n\pi/7)$

Think first. Reveal a hint when the class is ready.

2022 JM02