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Question Number 69873 by Masumsiddiqui399@gmail.com last updated on 28/Sep/19

Commented by mathmax by abdo last updated on 28/Sep/19

$$P_{n-1}=\prod _{k=1}^{n-1}(1-z_k)=\prod _{k=1}^{n-1}(1-e^{\frac{i2k\pi }{n}})$$$$=\prod _{k=1}^{n-1}(1-cos\left(\frac{2k\pi }{n}\right)-isin\left(\frac{2k\pi }{n}\right))$$$$=\prod _{k=1}^{n-1}(2{sin}^2\left(\frac{k\pi }{n}\right)-2isin\left(\frac{k\pi }{n}\right)cos\left(\frac{k\pi }{n}\right))$$$$=\prod _{k=1}^{n-1}(-2isin\left(\frac{k\pi }{n}\right))e^{\frac{ik\pi }{n}}={(-2i)}^{n-1}\prod _{k=1}^{n-1}sin\left(\frac{k\pi }{n}\right)e^{\frac{i\pi }{n}\sum _{k=1}^{n-1}k}$$$$={(-2i)}^{n-1}\prod _{k=1}^{n-1}sin\left(\frac{k\pi }{n}\right)e^{\frac{i\pi }{n}\frac{(n-1)n}{2}}$$$$={(-2i)}^{n-1}\prod _{k=1}^{n-1}sin\left(\frac{k\pi }{n}\right){\left(i\right)}^{n-1}$$$$=2^{n-1}\prod _{k=1}^{n-1}sin\left(\frac{k\pi }{n}\right)$$

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