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If cos⁻¹ x - cos⁻¹ (y/2) = α, where -1 ≤ x ≤ 1, -2 ≤ y ≤ 2, x ≤ y/2, then for all x, y, 4x² - 4xy cos α + y² is equal to:

1. 4 sin² α
2. 2 sin² α
3. 4 sin² α - 2x²y²
4. 4 cos² α + 2x²y²

Correct answer: 4 sin² α

Solution

The expression 4x² - 4xy cos α + y² can be rewritten using the identity for the cosine of the angle difference, leading to the conclusion that it simplifies to 4 sin² α, which is consistent with the relationship between the angles and their sine values.

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