Solving without induction

Question

$$\prod_{k=1}^n\cos\frac{x}{2^k}=\frac{\sin{x}}{2^n\sin\frac{x}{2^n}}$$ I tried to prove this without induction, but I can't come up with any idea. My teacher solved it with induction, which is the easy way, but she suggested us to try to solve it using a trigonometric trick. Could you please give me a hint?

Answer 1

Hint: With the double angle formula, one can show that

$$\cos\left(\frac{x}{2^n}\right) = \frac{1}{2}\cdot\frac{\sin\left(\frac{x}{2^{n-1}}\right)}{\sin\left(\frac{x}{2^n}\right)}$$

If you multiply consecutive terms of the LHS, some cancellation on the RHS occurs to give the desired identity.

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(I confess taking this from here.)