1.

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

Answer» 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,
`=1/((2^64)sintheta)(sin(2^64)theta)`
Putting value of `theta`, our expression becomes,
`=1/((2^64)sin((2pi)/(2^64-1) ))(sin(2^64)((2pi)/(2^64-1) ))`
`=1/((2^64)sin((2pi)/(2^64-1) ))(sin((2pi)/(2^64-1) ))`
`=1/(2^64)`
`=1/((2^4)^16) = 1/16^16`
So, value of the given expression will be `1/16^16`.


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