 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2013/10/b/solution

s = sin θ

and

c = cos θ

sin 8 θ = 8 c^{7} s - 56 c^{5} s^{3} + 56 c^{3} s^{5} - 8 c s^{7} = sc [8 c^{6} - 56 c^{4} s^{2} + 56 c^{2} s^{4} - 8 s^{6}] .

Substitute

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

and collect powers:

sin 8 θ = sc [8 - 80 s^{2} + 192 s^{4} - 128 s^{6}] .

Therefore the requested leading coefficient is

a = - 128.

The other coefficients, if retained as an algebraic check, are

b = 192, c = - 80, d = 8

. No division by

s

c

is needed, so the identity remains valid when either vanishes.

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