Your expression $ \frac{\sin x}{x} = \frac{\sin \frac{x}{2^k}}{ \frac{x}{2^k}} \prod_{i=1}^{k} \cos \frac{x}{2^i} $ is correct. Maybe we should separate out the very special case $x=0$, and from then on assume that $x\ne 0$. For $x=0$, $\frac{\sin x}{x}$ is formally undefined, but it is natural to set it equal to $1$. Then the formula works. If we are feeling in a very formal mood, we should prove the correctness of your expression by induction on $k$. However, I think that would be overkill.
We want to find the limit of the expression on the right as $k\to \infty$. As you observed, $\frac{x}{2^k}\to 0$ as $k\to \infty$. That certainly does not need proof. However, it is necessary to observe that since $\lim_{t=0}\frac{\sin t}{t}=1,$ we have $\lim_{k\to\infty} \frac{\sin \frac{x}{2^k}}{ \frac{x}{2^k}}=1.$ There will unfortunately be some special cases that require special treatment. We deal first with the "general" case when $x$ is not an integer multiple of $\pi$. Then your expression can be rewritten as $\frac{\sin x}{x}\frac{\frac{x}{2^k}}{\sin\frac{x}{2^k}}=\prod_{i=1}^{k} \cos \frac{x}{2^i}.$ Since $\lim_{k\to\infty}\frac{\sin x}{x}\frac{\frac{x}{2^k}}{\sin\frac{x}{2^k}}$ exists and is equal to $\frac{\sin x}{x}$, we conclude that $\lim_{k\to\infty} \prod_{i=1}^{k} \cos \frac{x}{2^i}$ also exists and is equal to $\frac{\sin x}{x}$.
If $x\ne 0$ is an integer multiple of $\pi$, we have to choose $k$ large enough so that $\sin\frac{x}{2^k}\ne 0$, else when we rewrite your expression, we might be dividing by $0$. Minor point. For such $x\ne 0$, $\frac{\sin x}{x}=0$, and one of the cosines is $0$. So, in that case also, apart from a technicality discussed below, the formula looks correct.
Technical remark: In the formal definition of an infinite product, we say that $\prod_{i=1}^\infty a_i$ converges if $\lim_{k\to\infty}\prod_{i=1}^k a_i$ exists and is not equal to $0$. So technically when $x$ is an integer multiple of $\pi$, the infinite product does not converge!