Correct Answer - `(3+sqrt(5))/(2)`
`(tan9^(@)+cot9^(@))/(tan27^(@)+cot27^(@))=(tan9^(@)+(1)/(tan9^(@)))/(tan27^(@)+(1)/(tan27^(@)))`
`=(Tan^(2)9^(@)+1)/(2 tan 9^(@))(2tan 27^(@))/((tan^(2)27^(@)+1)`
`=(sin 54^(@))/(sin18^(@))=(cos 36^(@))/(sin18^(@))`
`=(sqrt(5)+1)/(4).(4)/(sqrt(5)-1)`
`=(sqrt(5)+1)/(sqrt(5)-1)xx(sqrt(5)+1)/(sqrt(5)+1)=(3+sqrt(5))/(2)`.
