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Compute the product (2 - 2cos(2*pi*1/13)) * (2 - 2cos(2*pi*2/13)) *…
Question
Compute the product (2 - 2cos(2*pi*1/13)) * (2 - 2cos(2*pi*2/13)) * ... * (2 - 2cos(2*pi*12/13)), the product taken over k = 1, 2, ..., 12. Report the exact integer value.
✓ Verified answer: 169checked by our engine — not a guess
Step-by-step solution
Let w = e^{2*pi*i/13}, so the non-trivial 13th roots of unity are w^k for k = 1,...,12.
Note 2 - 2cos(2*pi*k/13) = (1 - w^k)(1 - w^{-k}) = |1 - w^k|^2.
Therefore the product equals prod_{k=1}^{12} |1 - w^k|^2 = (prod_{k=1}^{12} |1 - w^k|)^2.
Now prod_{k=1}^{n-1}(x - w^k) = (x^n - 1)/(x - 1) = 1 + x + ...
+ x^{n-1}; evaluating at x = 1 gives prod_{k=1}^{n-1}(1 - w^k) = n.
For n = 13 this product equals 13 (real and positive), so the product of moduli is 13 and the answer is 13^2 = 169.
Final answer169
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