 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/7/a/solution

Put

θ = π /12

cos^{2} (2 θ) = 1 - sin^{2} (2 θ)

. Since

2 θ = π /6

is in the first quadrant, the positive square root is required. The half-angle identities then give the requested expression in terms of

sin (π /6)

sin^{2} \frac{π}{12} = \frac{1 - 1 - sin ^{2} ( π /6 )}{2} .

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