Trigo or algebra?

Calculus Level 3

sin ⁡ ( x ) = ∏ n = 0 ∞ ( x 2 − n 2 π 2 ) \sin(x) = \displaystyle\prod_{n=0}^{\infty} (x^2-{n^2\pi^2})

  1. Are there an infinite number of solutions to the above equation?
  2. Is the equation true for all values of x x ?
(1)Yes (2)Yes (1) Yes (2) No (1)No (2) No

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Akash Shukla by Akash Shukla , 26, India

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1 solution

For m in N, x = m*pi, at least one of the factors is null on the right, and sin(x)=0. Then there is an infinite number of solution. But for x= pi/sqrt(2) all the factors are strictly superior to 1 (in absolute value), and even superior or egal to x, so necessarily the product is strictly superior to 1 and cannot be equal to sin(x) that is inferior or egal to 1, so the equation does not hold true for all values of x.

(Sorry for my eventual language mistakes)

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