If Z= [tan 2x -i(sin x+cos x)]/[1+i sin x] is completely imaginary, find x.(0<=x<=(pi/2) )

Question

I understand that here we have to equate the real part to 0,but how do we isolate the real part.I tried multi plying both numerator and denominator by (1-2isinx),but it doesn't seem to be leading me any further.

Answer 1

"real"ize the denominator by multiplying top and bottom by 1-i sinx. Now you have

[tan 2x - i(sinx+cosx)][1 - i sinx]/[(1-i sinx)(1+i sinx)]

[tan2x - (sinx+cosx)(sinx) + i(junk)]/(1+sin^2 x)

to get zero real part, we need

tan2x = sin^2x + sinx cosx
any multiple of π clearly satisfies this equation.

It also makes Z have zero real part.

Answer 2

Thank you very much Steve!