Weighted Sums and Triangular Arrays

Prove the identity \[ \frac {\sin \theta }{\theta } = \int _0^1 e^{i\theta (2x-1)}\,dx. \] Then use this to show that \[ \frac {\sin \theta }{\theta } = \prod _{n=1}^{\infty }\cos (\theta /2^n). \] Hint: Use that a Uniform\((0,1)\) random variable can be written as an infinite weighted sum of iid Bernoulli\((1/2)\) random variables.