JEE Advanced 2016 Paper 1 · Mathematics · Complex Numbers · hard

Let ω=e^2π i/7. The value of Σk=1^6(1)/(1-cos dfrac2π k7) is

Numerical answer

Let

\omega=e^{2\pi i/7}

. The value of

\displaystyle\sum_{k=1}^{6}\frac{1}{1-\cos\dfrac{2\pi k}{7}}

Show answer and solution

Answer: 8

Since

1-\cos\theta=2\sin^2(\theta/2)

, the sum equals

\frac12\sum_{k=1}^{6}\csc^2\frac{k\pi}{7}

. To evaluate it, use the roots of unity:

\left|1-\omega^k\right|^2=2-2\cos\frac{2\pi k}{7}=4\sin^2\frac{k\pi}7

, so the sum is

2\sum_{k=1}^{6}\frac{1}{|1-\omega^k|^2}

. The numbers

\frac1{1-\omega^k}=\frac12+\frac i2\cot\frac{k\pi}{7}

are the roots of a degree six polynomial obtained from

(1+1/w)^7=1

type relations, and the standard result is

\sum_{k=1}^{n-1}\csc^2\frac{k\pi}{n}=\frac{n^2-1}{3}

. With

n=7

this equals

16

, so the required sum is

\tfrac12\cdot16=8

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