infinite product – Prove the formula for $ sinx / x $ using the power series

I would appreciate some ideas for the following:

"Prove it $ frac { sin {x}} {x} = prod_ {n = 1} ^ { infty} cos { frac {x} {2 ^ n}} $ using power series. "

I am aware that this identity can be shown with the help of trigonometric identities and a telescopic product. In addition, you can get some other proof using the infinite product expressions $ sin {x} = x prod_ {k = 1} ^ { infty} (1- frac {x ^ 2} {k ^ 2 pi ^ 2}) $ and $ cos {x} = prod_ {k = 1, k text {odd}} ^ infty (1- frac {4x ^ 2} {k ^ 2 pi ^ 2}) $.

However, since the question explicitly mentions the power series, I wondered if there is any proof that directly uses the power series? I've tried calculating derivatives and coefficients, but they seem to be quite unpleasant.