# How to simply: <munderover> &#x220F;<!-- ∏ --> <mrow class="MJX-TeXAtom-ORD"> k

How to simply: $\prod _{k=1}^{100}\left(1+2cos\frac{2\pi {.3}^{k}}{{3}^{100}+1}\right)$?
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plodno8n
HINT:
For $\mathrm{sin}y\ne 0,$,
$1+2\mathrm{cos}2y=1+2\left(1-2{\mathrm{sin}}^{2}y\right)=\frac{\mathrm{sin}3y}{\mathrm{sin}y}$
Observe the Telescoping nature and use $\mathrm{sin}\left(\pi -u\right)=+\mathrm{sin}u$

Mohamed Mooney
$1+2\mathrm{cos}2{y}_{k}=1+2\left(1-2{\mathrm{sin}}^{2}{y}_{k}\right)=\frac{\mathrm{sin}3{y}_{k}}{\mathrm{sin}{y}_{k}}$
${y}_{k}=\frac{{3}^{k}\pi }{{3}^{100}+1}$
Product will reduce to
$\frac{\mathrm{sin}3{y}_{100}}{\mathrm{sin}{y}_{1}}$
Just observe
$\frac{\mathrm{sin}\left(3\pi -{y}_{1}\right)}{\mathrm{sin}{y}_{1}}$