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I am trying to factorize $sin(x)\over \sin(x)\over x$ which by Taylor series expansion and using the roots is $a $a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} \right) \left(1 + \frac{x}{2\pi} \right) \left(1 - \frac{x}{3\pi} \right) \left(1 + \frac{x}{3\pi} \right) \cdot ...$cdots$$

Now I was told that this nasty factor $a$ conveniently becomes $1$ due to Weierstrass’s Factorization Theorem which is a transcendental generalization of the Fundamental Theorem of Algebra.

My question
Could you please show me how $a$ is being neutralized using this theorem? Or don't you even need this theorem to do so?
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Using Weierstrass’s Factorization Theorem

I am trying to factorize $sin(x)\over x$ which by Taylor series expansion and using the roots is $a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} \right) \left(1 + \frac{x}{2\pi} \right) \left(1 - \frac{x}{3\pi} \right) \left(1 + \frac{x}{3\pi} \right) \cdot ...$

Now I was told that this nasty factor $a$ conveniently becomes $1$ due to Weierstrass’s Factorization Theorem which is a transcendental generalization of the Fundamental Theorem of Algebra.

My question
Could you please show me how $a$ is being neutralized using this theorem? Or don't you even need this theorem to do so?