Use the preceding identities to evaluate the cosine-square sum.

01 / Standard#Your turn

\cos^2\frac{2\pi}7+\cos^2\frac{4\pi}7+\cos^2\frac{6\pi}7

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Square (ωⁿ+ω⁻ⁿ)/2 for n=1,2,3.

Hint 2

Reduce negative exponents modulo seven.

Worked solution

Expand each squared cosine.
$S=\frac14\sum_{n=1}^3(\omega^{2n}+2+\omega^{-2n})$
The six nonconstant powers are exactly ω,…,ω⁶ in a different order.
$S=\frac14(6+\omega+\omega^2+\cdots+\omega^6)=\frac14(6-1)=\frac54$

5/4.

Checks and common pitfalls: The square creates a constant 2 in each conjugate product.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Expand each squared cosine.
$S=\frac14\sum_{n=1}^3(\omega^{2n}+2+\omega^{-2n})$
The six nonconstant powers are exactly ω,…,ω⁶ in a different order.
$S=\frac14(6+\omega+\omega^2+\cdots+\omega^6)=\frac14(6-1)=\frac54$

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

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