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Evaluate the product (1 + cos(π/10))(1 + cos(3π/10))(1 + cos(7π/10))(1 + cos(9π/10)).

1. 1/8
2. 1/16
3. 1/32
4. None of these

Correct answer: 1/16

Solution

Pair the terms: cos(7π/10)=-cos(3π/10) and cos(9π/10)=-cos(π/10). The product becomes (1-cos^2(π/10))(1-cos^2(3π/10))=sin^2(π/10)sin^2(3π/10). Using sin(π/10)sin(3π/10)=1/4 gives (1/4)^2=1/16.

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