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Prove that cos theta* cos 2theta* cos2^...

Prove that `cos theta* cos 2theta* cos2^(2)theta * cos2^(3)theta x... cos 2^(n-1) theta = 1/2^(n)` , when `theta = pi/(2^(n)+1)`.

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Using induction,prove that cos theta cos2 theta cos2^(2)theta...cos2^(n-1)theta=(sin2^(n)theta)/(2^(n)sin theta)

Knowledge Check

  • Find the value of cos theta. cos 2theta. cos 2^(2)theta . cos 2^(3)theta"……." cos 2^(n-1) theta is

    A
    `(cos2^(n)theta)/(2^(n)sintheta)`
    B
    `(sin2^(n)theta)/(2^(n)costheta)`
    C
    `(cos2^(n)theta)/(2^(n)costheta)`
    D
    `(sin2^(n)theta)/(2^(n)sin theta)`

  • For all n in N, cos theta cos 2theta cos 4theta ....cos2^(n-1)theta equals to

    A
    `(sin2^(n)theta)/(2^(n)sintheta)`
    B
    `(sin2^(n)theta)/(sintheta)`
    C
    `(cos2^(n)theta)/(2^(n)cos2theta)`
    D
    `(cos2^(n)theta)/(2^(n)sin2theta)`

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    If (2^(n)+1)theta=pi then 2^(n)cos theta cos2 theta cos2^(2)theta......cos2^(n-1)theta=

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