To prove $\frac{\sin x}x=\cos(x/2)\cos(x/4)\cos(x/8)\dots$

Question

I inverted the LHS such that first term is $\cos(x/2^n)$ then and multiplied it by $2\sin(x/2^n)/2\sin(x/2^n)$ and evaluated it by $\sin2A=2\sin A\cos A$ but now I am stuck. Are any other ways to prove it? Please give suggestions. What should I do?

Answer 1

Prove by induction that:

$$ \prod_{i=1}^n\cos\frac{x}{2^i}=\frac{\sin x}{2^n\sin{\frac{x}{2^{n}}}}, $$

and let $n\rightarrow\infty$.

Answer 2

Expand again and again $\sin(x)$ by $\sin(x)=2\cos(\frac{x}{2})\sin(\frac{x}{2})$ and so on, you have $$\frac{\sin x}{x} = \cos\left(\frac{x}{2}\right)\cos\left(\frac{x}{4}\right)\cos\left(\frac{x}{8}\right)\cos\left(\frac{x}{16}\right)\dots\cos\left(\frac{x}{2^n}\right)\frac{\sin(\frac{x}{2^n})}{\frac{x}{2^n}}$$ and you let $n$ goes to infinity.