1963 IMO Problems/Problem 5

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Problem

Prove that .

Solution

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We have S = cos(π/7) - cos(2π/7) + cos(3π/7) = cos(π/7) + cos(3π/7) + cos(5π/7)

Then, by product-sum formulae, we have S * 2* sin(π/7) = sin(2π/7) + sin(4π/7) - sin(2π/7) + sin(6π/7) - sin(4π/7) = sin(6π/7) = sin(π/7)

Thus S = 1/2

See Also