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I came across this question in the following form: Compute the following infinite product $$\left[\sin (x)\cos \left(\frac{x}{2}\right)\right]^{1/2}\cdot \left[\sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{4}\right)\right]^{1/4}\cdot \left[\sin \left(\frac{x}{4}\right) \cos \left(\frac{x}{8...