If - -2n-1- then cos-cos 2- 2cos 22- -cos 2n-1- -1-an- Find a

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Sum of Coefficients of All Terms

If θ =π/2n+...

Question

If $θ = \frac{π}{2^{n} + 1}$ , then $cos θ cos 2 θ 2 cos 2^{2} θ . . . cos 2^{n - 1} θ = \frac{1}{a^{n}}$ . Find $a$

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Solution

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θ = \frac{π}{2^{n} + 1}

2^{n} cos θ cos 2 θ cos 2^{2} θ . . . . cos 2^{n - 1} θ = 1

cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{4 π}{7} = - \frac{1}{8}

cos θ cos 2 θ cos 4 θ . . . cos 2^{n - 1} θ = - \frac{1}{2^{n}}

θ = \frac{π}{2^{n} - 1}

cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{4 π}{7} = \frac{1}{8}

cos θ cos 2 θ cos 4 θ . . . cos 2^{n - 1} θ = \frac{1}{2^{n}}

θ = \frac{π}{2^{n} - 1}

(2^{n} + 1) θ = π,

2^{n} \cos θ \cos 2 θ \cos 2^{2} θ . . . \cos 2^{n - 1} θ = 1

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If $(2^{n} + 1) θ = π, then 2^{n} c o s θ c o s 2 θ \dots c o s 2^{2} θ \dots 2^{n - 1} θ =$

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