1.

If `z+1//z=2costheta,`prove that `|(z^(2n)-1)//(z^(2n)+1)|=|tanntheta|`

Answer» `z+(1)/(2)=2costheta`
or `z^(2)-2costhetaz+1=0`
or `z=(2costheta+-sqrt(4cos^(2)theta-4))/(2)`
`=costheta+-isintheta`
Taking positive sign, we get
`z=costheta+isintheta`
`:." "(1)/(z)=(costheta-isintheta)`
`:." "(z^(2n)-1)/(z^(2n)+1)=(z^(n)-(1)/(z^(n)))/(z^(n)+(1)/(z^(n)))`
`=((costheta+isintheta)^(n)-(costheta-isintheta)^(n))/((costheta+isintheta)^(n)+(costheta-isintheta)^(n))`
`=(2isinntheta)/(2cosntheta)`
`=itan ntheta`
Taking negative sign, we get
`(z^(2n)-1)/(z^(2n)+1)=(-2isinntheta)/(2costheta)=-itanntheta`
`implies|(z^(2n)-1)/(z^(2n)+1)|=|+-itantheta|=tanntheta`


Discussion

No Comment Found

Related InterviewSolutions