 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ia/paper-1/8a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

m

(cos θ + i sin θ)^{m} = cos (m θ) + i sin (m θ)

. Taking the real part for

m = 5

gives

cos 5 θ = cos^{5} θ - 10 cos^{3} θ sin^{2} θ + 5 cos θ sin^{4} θ .

Replace

sin^{2} θ

1 - cos^{2} θ

and expand:

cos 5 θ = 16 cos^{5} θ - 20 cos^{3} θ + 5 cos θ .

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