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For the determinant | 1 a a^2 | | cos((n−1)x) cos(nx) cos((n+1)x) | | sin((n−1)x) sin(nx) sin((n+1)x) | its value becomes zero when

1. sin x = 0
2. cos x = 0
3. a = 0
4. cos x = (1+a^2)/(2a)

Correct answer: cos x = 0

Solution

The determinant becomes zero when the second and third rows are linearly dependent, which occurs when cos x = 0, as this leads to the sine and cosine functions being orthogonal and thus creating a situation where the rows can be expressed as multiples of each other.

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