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Trigonometry

I have just realized that I haven’t posted a good puzzle here in a long time, so here it goes one on Trigonometry, that the average student of Calculus should be able to tackle: you can use anything you think it could help: derivatives, symmetry, periodicity, integration, summation, go to several variables, differential equations, etc

Prove that, for all real values x \in \mathbb{R}, it is

\sin x = x \cos\big(\tfrac{x}{2}\big) \cos\big(\tfrac{x}{4}\big) \cos\big(\tfrac{x}{6}\big) \cos\big(\tfrac{x}{8}\big) \dotsb \cos\big( \tfrac{x}{2n}\big) \dotsb

or, in a more compact notation,

\sin x = x \displaystyle{\prod_{n=1}^\infty \cos \big( \tfrac{x}{2n} \big)}

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