Question

If $S = cos^{2}\frac{\pi}{n} + cos^{2}\frac{2\pi}{n} + \ldots\ldots\ldots\ldots\ldots + cos^{2}\frac{(n - 1)\pi}{n},$ then S equals

• A. $\frac{n}{2}(n + 1)$
• B. $\frac{1}{2}(n - 1)$
• C. $\frac{1}{2}(n - 2)$
• D. $\frac{n}{2}$

Answer 1

Answer: (C)

$\frac{1}{2}(n - 2)$

Answer 2

$\frac{1}{2}\left\lbrack 1 + \cos\frac{2\pi}{n} + 1 + \cos\frac{4\pi}{n} + 1 + \cos\frac{6\pi}{n} + \ldots + 1 + \cos 2(n - 1)\frac{\pi}{n} \right\rbrack$=$\frac{1}{2}\left\lbrack n - 1 + \sum_{k = 1}^{n - 1\sum\frac{2k\pi}{n}}\cos\lbrack\rbrack \right\rbrack$= $\frac{1}{2}\lbrack n - 1 - 1\rbrack = \frac{1}{2}(n - 2)$