Evaluating limit $\lim_{k\to \infty}\prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}$

Question

I stumbled across the following question which asked to evaluate...

$$\lim_{k\to \infty}\prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}$$ I at first tried writing few terms $$\cos{\left(\frac {x}{2}\right)}\cos{\left(\frac {x}{4}\right)}\cos{\left(\frac {x}{8}\right)}...$$ I used the Half-angle formula to write$$\cos{\left(\frac {x}{2}\right)}=\pm\sqrt{\frac{1+\cos(x)}{2}}$$ Therefore, $$\sqrt{\frac{1+\cos(x)}{2}}\sqrt{\frac{1+\sqrt{\frac{1+\cos(x)}{2}}}{2}}...$$ As there are infinitely many two's in the denominator, the denominator goes to $\infty$ which means $$\lim_{k\to \infty}\prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}=0$$

So..My question is ...Am I correct?... If not, Could you please give me some hint to how should I proceed ?

Answer 1

Hint. One may use$$ \cos{\left(\frac {x}{2^r}\right)}=\frac12 \cdot \frac{\sin{\left(\frac {x}{2^{r-1}}\right)}}{\sin{\left(\frac {x}{2^r}\right)}} $$ giving, by a telescoping product, $$ \prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}=\frac1{2^k}\cdot\prod_{r=1}^k\frac{\sin{\left(\frac {x}{2^{r-1}}\right)}}{\sin{\left(\frac {x}{2^r}\right)}}=\frac1{2^k}\cdot\frac{\sin{x}}{\sin{\frac {x}{2^k}}}=\frac{\large\frac{\sin x}x}{\large\frac{\sin{\frac {x}{2^k}}}{\frac {x}{2^k}}} $$ then let $k \to \infty$.

Can you take it from here?

Answer 2

Check by induction that $$ \sin(x) = 2^m\sin(2^{-m}x)\prod_{j=1}^m \cos(2^{-j} x)$$ since $$ \sin(x)=2^{}\sin(2^{-1}x)\cos(2^{-1}x)=2^{2}\cos(2^{-1}x)\cos(2^{-2}x)\sin(2^{-2}x)=.......$$ Then, $$\lim_{m\to \infty } \prod_{j=1}^m \cos(2^{-j} x) = \lim_{m\to -\infty } \frac{2^{-m}x}{\sin(2^{-m}x)} \frac{\sin(x)}{x} = \frac{\sin(x)}{x}$$