Gheorghe ANDREI - COMPLEX NUMBERS (part II) - Some Problems -- 5 / Gheorghe ANDREI - NUMERE COMPLEXE (partea II) - Câteva Probleme -- 5 / Gheorghe ANDREI – LICZBY ZŁOŻONE (część II) – Niektóre problemy – 5

 I wrote about the book here. 

It's not for nothing that I wrote somewhere that THIS BOOK IS FULL OF SCRATCHES. (Maybe that's why the author withdrew Part I from sale)

                    We are now debating Problem 62***, page 157 (solved on pages 516-519).

       

                     " 62***.  Calculate  $P_n\; Q_n\;and\; R_n$, where  $n\in \mathbb{N},\;n\geqslant 3,\;a\in\mathbb{R}^*\;:$

 $$a)\;\;\;P_n=\prod_{k=0}^{n-1} \cos \left (a+\frac{k\pi}{n}\right )\;;$$

$$b)\;\;Q_n=\prod_{k=0}^{n-1}\sin \left (a+\frac{k\pi}{n}\right )\;;$$

$$c)\;\;R_n=\prod_{k=0}^{n-1}\tan \left (a+\frac{k\pi}{n}\right )."$$

ANSWER

$a)\;\;P_n=$

$b)\;\;Q_n=\frac{\sin nx}{2^{n-1}}$

$c)\;\;R_n=$

ghjgjg

(in construction)

     Remark CiP   On page 518 the following stupid answer is given:

$$Q_n=\begin{cases}\frac{(-1)^n\cdot \sin na}{2^{n-1}}\;\;\;if\;\;n-even\\\frac{(-1)^{n+1}\cdot \sin na}{2^{n-1}}\;\;\;if\;\;n-odd\end{cases} \tag{Q}$$