 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/3/solution

sin^{4} θ + cos^{4} θ = (sin^{2} θ + cos^{2} θ)^{2} - 2 sin^{2} θ cos^{2} θ = 1 - \frac{1}{2} sin^{2} (2 θ) .

The required value

1/2

therefore occurs when

sin^{2} (2 θ) = 1

. Within

0 \leq θ \leq π

, the only possibilities are

2 θ = π /2

and

3 π /2

. Hence

θ = \frac{π}{4} or \frac{3 π}{4} .

Equivalently equality requires

sin^{2} θ = cos^{2} θ = 1/2

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