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Math Help - Vieta's Formula

  1. #1
    Guy
    Guy is offline
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    Vieta's Formula

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

    \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 \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 t = \frac \pi 2 makes sense, but I end up with needing to show

    \displaystyle<br />
\cos \left(\frac \pi {2^{k + 1}}\right) = \sqrt{\frac {1 + \cos \frac \pi {2^k}} 2}<br />

    to get the answer. So far I haven't been successful at this. Any points in the right direction would be appreciated.
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  2. #2
    Guy
    Guy is offline
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    Never mind, I'm stupid. Just use the formula for \cos^2 (x).
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