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Cos { 2 pi/2^64 - 1) cos{2^2 pi/(2^64-1)...

` Cos { 2 pi/2^64 - 1) cos{2^2 pi/(2^64-1)}.........cos{2^64 pi/(2^64 - 1)}` =

Text Solution

Verified by Experts

Let `(2pi)/(2^64-1) = theta`
Then, our expression becomes,
`costhetacos2thetacos4theta......cos(2^63)theta`
`=1/(2sintheta)(2sinthetacosthetacos2thetacos4theta.....cos(2^63)theta)`
`=1/(2sintheta)(sin2thetacos2thetacos4theta.....cos(2^63)theta)`
`=1/(4sintheta)(2sin2thetacos2thetacos4theta.....cos(2^63)theta)`
`=1/(4sintheta)(sin4thetacos4theta.....cos(2^63)theta)`
Similarly, if we continue, we will get the required expression as,
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