 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/12/d/solution

(cos θ + i sin θ)^{4} = cos 4 θ + i sin 4 θ

. Take real parts of the binomial expansion:

cos 4 θ = cos^{4} θ - 6 cos^{2} θ sin^{2} θ + sin^{4} θ .

Eliminate

sin^{2} θ

using

1 - cos^{2} θ

and collect powers:

cos 4 θ = 8 cos^{4} θ - 8 cos^{2} θ + 1.

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