Question

The value of cos2x for which the expression 1+ icosx/1-2icosx where( i root -1) is purely imaginary is

Answer 1

Given :  (1  + i Cosx)/(1 - 2iCosx)   is purely imaginary

To find  :  value of cos2x

Solution:

(1  + i Cosx)/(1 - 2iCosx)

= (1 + i Cosx) (1 + 2iCosx) / ( 1 - 2i Cosx )(1 + 2 i Cosx)

=(1  + 2i²Cos²x  + iCosx + 2iCosx) / ( 1  - 4i²Cos²x)

= ( 1 - 2Cos²x + 3iCosx) / ( 1 + 4 Cos²x)

= (1 - 2Cos²x)/ ( 1 + 4 Cos²x)   + i 3Cosx/( 1 + 4 Cos²x)

Puerly imaginary  hence

(1 - 2Cos²x)/ ( 1 + 4 Cos²x) = 0

=> 1 - 2Cos²x   =  0  

=>  -  Cos2x  =  0

=>   Cos2x  = 0

value of cos2x  = 0 for which (1  + i Cosx)/(1 - 2iCosx)  is purely imaginary

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