 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/11/c/solution

cos 5 θ

is the real part of

(cos θ + i sin θ)^{5}

c = cos θ

and

s = sin θ

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

s^{2} = 1 - c^{2}

and expand:

c^{5} - 10 c^{3} (1 - c^{2}) + 5 c (1 - 2 c^{2} + c^{4}) = 16 c^{5} - 20 c^{3} + 5 c .

Therefore

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

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