 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2019/9/a/solution

Put

w = e^{i θ}

w - w^{- 1} = 2 i sin θ

(w - w^{- 1})^{7} = w^{7} - 7 w^{5} + 21 w^{3} - 35 w + 35 w^{- 1} - 21 w^{- 3} + 7 w^{- 5} - w^{- 7} .

Group opposite powers and use

w^{j} - w^{- j} = 2 i sin (j θ)

(2 i)^{7} sin^{7} θ = 2 i [sin 7 θ - 7 sin 5 θ + 21 sin 3 θ - 35 sin θ] .

Since

(2 i)^{7} = - 128 i

- 2 i

yields

64 sin^{7} θ = 35 sin θ - 21 sin 3 θ + 7 sin 5 θ - sin 7 θ .

This derivation is an application of the same exponential representation underlying De Moivre's theorem.

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