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$\sum_{r = 1}^{n - 1} \cos^{2} \frac{r π}{n}$ is equal to

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a

n2

b

n−12

c

n2−1

d

none of these

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detailed solution

Correct option is C

We have,∑r=1n−1 cos2⁡rπn=12∑r=1n−1 1+cos⁡2rπn=12∑r=1n−1 1+12∑r=1n−1 cos⁡2rπn=(n−1)2+12cos⁡2πn+cos⁡4πn+…+cos⁡2(n−1)πn=(n−1)2+12×cos⁡2πn+(n−2)πnsin⁡(n−1)πnsin⁡π=(n−1)2+12cos⁡π=n−12−12=n2−1

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