Anybody know a proof of $\prod_{n=1}^\infty\cos(x/2^n)=\sin x/x$.

Using the trig identity

$$\sin (2t) = 2\sin (t) \cos (t),$$

we have

$$\prod_{n = 1}^N \cos(x/2^n) = \prod_{n = 1}^N \frac{\sin(x/2^{n-1})}{2\sin(x/2^n)} = \frac{\sin(x)}{2^N\sin(x/2^N)} = \frac{\sin x}{x}\cdot \frac{x/2^N}{\sin(x/2^N)}$$

Take the limit as $N \to \infty$ and use the fact $\lim_{t\to 0} \frac{\sin t}{t} = 1$ to obtain the result.

Hint

$$\cos(x/2^n)=\frac12\frac{\sin(x/2^{n-1})}{\sin(x/2^n)}$$ and telescope.