JEEOlympiad
Compute the exact value of cot²(π/13) + cot²(2π/13) + cot²(3π/13) +…
Question
Compute the exact value of cot²(π/13) + cot²(2π/13) + cot²(3π/13) + cot²(4π/13) + cot²(5π/13) + cot²(6π/13). Give a single integer.
✓ Verified answer: 22checked by our engine — not a guess
Step-by-step solution
There is a classical identity: for a positive integer n, Σ (k=1..n) cot²(kπ/(2n+1)) = n(2n−1)/3.
This follows from the fact that the numbers cot²(kπ/(2n+1)), k = 1..n, are the roots of a polynomial of degree n derived from expanding sin((2n+1)θ) in powers of cot θ; the sum of roots equals (by Vieta) the ratio of the appropriate coefficients, n(2n−1)/3.
Here 2n + 1 = 13, so n = 6. The sum equals 6·(2·6 − 1)/3 = 6·11/3 = 22.
Final answer22
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