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Prove that : (1-cos theta+i sin theta)/(...

Prove that : `(1-cos theta+i sin theta)/(1+i sin theta +cos theta)=ie^-1 tan (theta/2)`

Text Solution

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LHS
`(1-1+2sin^2theta/2+i2sintheta/2costheta/2)/(1+2cos^22theta/2-1+i2sintheta/2costheta/2)`
`(i2sintheta/2(isintheta/2+costheta/2))/(2costheta/2(costheta/2+isintheta/2))`
`itantheta/2(costheta/2-isintheta/2)/(costheta/2+isintheta/2)`
`itantheta/2(e^(-itheta/2))/(e^(itheta/2)`
`itanthetae^(-itheta)`.
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