Table of Integrals, Series, and Products, 8th Edition

1.39 The representation of cosines and sines of multiples of the angle as finite Products

1.391

1. 

$\begin{array}{cc}sinnx=nsinxcosx\underset{k=1}{\overset{\frac{n-2}{2}}{\mathrm{\Pi }}}\left(1-\frac{{sin}^{2}x}{{sin}^2\frac{k\pi }{n}}\right)& [n\text{is even}]\end{array}$

  JO (568)

2. 

$\begin{array}{cc}cosnx=\underset{k=1}{\overset{\frac{n}{2}}{\mathrm{\Pi }}}\left(1-\frac{{sin}^{2}x}{{sin}^2\frac{\left(2k-1\right)\pi }{2n}}\right)& [n\text{is even}]\end{array}$

  JO (569)

3. 

$\begin{array}{cc}sinnx=nsinx\underset{k=1}{\overset{\frac{n-1}{2}}{\mathrm{\Pi }}}\left(1-\frac{{sin}^{2}x}{{sin}^2\frac{k\pi }{n}}\right)& [n\text{is odd}]\end{array}$

  JO (570)

4. 

$\begin{array}{cc}cosnx=cosx\underset{k=1}{\overset{\frac{n-1}{2}}{\mathrm{\Pi }}}\left(1-\frac{{sin}^{2}x}{{sin}^2\frac{\left(2k-1\right)\pi }{2n}}\right)& [n\text{is}\text{?}\text{odd}]\end{array}$

  JO (571)a ...