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

1.391

1.

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

JO (568)

2.

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

JO (569)

3.

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

JO (570)

4.

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

JO (571)a ...

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