1.

If `x_n`=cos`pi/(2^n)`+isin`pi/(2^n)`, prove that `x_1x_2x_3 x_oo=-1.`

Answer» Here, `x_n = cos(pi/(2^n))+isin(pi/(2^n))`
`:.L.H.S. = x_1x_2x_3...x_oo`
`= (cos(pi/(2))+isin(pi/(2)))(cos(pi/(2^2))+isin(pi/(2^2)))(cos(pi/(2^3))+isin(pi/(2^3)))...oo`
`=cos(pi/2+pi/2^2+pi/2^3+...oo)+isin(pi/2+pi/2^2+pi/2^3+...oo)`
`= cos((pi/2)/(1-1/2))+isin((pi/2)/(1-1/2))`
`=cospi+isinpi`
`= -1+0 = -1 = R.H.S.`


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