Hello all. I think this is the right forum for this. I am asked to show

$\displaystyle \frac 2 \pi = \frac{\sqrt 2}{2} \frac{\sqrt{2 + \sqrt 2}}{2} \frac{\sqrt{2 + \sqrt{2 + \sqrt 2}}}{2} \cdots$

first by showing $\displaystyle \frac {\sin t}{t} = \prod_{k = 1} ^ \infty \cos \frac t {2^k}$ (which I have done) and then presumably I choose an appropriate value of t to get the identity. Taking $\displaystyle t = \frac \pi 2$ makes sense, but I end up with needing to show

$\displaystyle \displaystyle

\cos \left(\frac \pi {2^{k + 1}}\right) = \sqrt{\frac {1 + \cos \frac \pi {2^k}} 2}

$

to get the answer. So far I haven't been successful at this. Any points in the right direction would be appreciated.