if the expression [sin(x/2) + cos(x/2) - itan(x/2)]/{1+i2sin(x/2)} is real, then the set of all possible values of x is _________

Dear raman

[sin(x/2) + cos(x/2) - itan(x/2)]/{1+i2sin(x/2)}

multiple 1-21sin(x/2)  in nominator and denomenator and equate coefficient of i to zero

-tanx/2 -2sinx/2(sinx/2 +cosx/2) =0

or sinx/2 ( 1+ 2cosx/2((sinx/2 +cosx/2)) =0

 or sinx/2(2+sinx/2 +cosx/2)=0

 term in the bracket can not equal to zero

so sinx/2 =0

 x/2 =n∏

x=2n∏  for n any intiger

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