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Question Number 158186 by mkam last updated on 31/Oct/21

Answered by puissant last updated on 31/Oct/21

$$sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+......$$$$\Rightarrow \frac{sinx}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+.....$$$$thesolutonsofthispolynomialare$$$$thesolutionofsinx(x=n\pi );(n\in \mathbb{Z})$$$$\Rightarrow \frac{sinx}{x}=(1+\frac{x}{\pi })(1-\frac{x}{\pi })(1+\frac{x}{2\pi })(1-\frac{x}{2\pi })..$$$$\Rightarrow \frac{sinx}{x}=(1-\frac{x^2}{{\pi }^2})(1-\frac{x^2}{4{\pi }^2})(1-\frac{x^2}{9{\pi }^2})...$$$$\Rightarrow \frac{sinx}{x}=\underset{k=1}{\prod ^{\mathrm{\infty }}}(1-\frac{x^2}{{\left(n\pi \right)}^2})...$$$$$$$$...........\mathcal{L}epuissant..........$$

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