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Prove that (cos alpha - cos beta)² + (sin alpha - sin beta)² = 4 sin²((alpha - beta)/2). Which statement is correct?

1. It is a valid identity
2. RHS should be 4 cos²((alpha-beta)/2)
3. RHS should be 2 sin²((alpha-beta)/2)
4. It holds only when alpha = beta

Correct answer: It is a valid identity

Solution

Expanding gives 2 - 2cos(alpha-beta) = 2(1 - cos(alpha-beta)) = 4 sin²((alpha-beta)/2).

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