Question

(Tan (theta)/1-cot (theta))+ (cot (theta)/1-tan (theta))

Answer 1

let theta = Ф
tanФ/(1-cotФ) + cotФ/(1-tanФ)
on resolving sin and cos 
=> sin²Ф/{cosФ(sinФ-cosФ)} - cos²Ф/{sinФ(sinФ-cosФ)}
=> {sin³Ф-cos³Ф}/{(sinФ-cosФ)sinФcosФ}
=> {(sinФ-cosФ)(sin²Ф+cos²Ф+cosФsinФ)}/{cosФsinФ(sinФ-cosФ)}
=> (1+cosФsinФ)/(cosФsinФ)                                                           (sin²Ф+cos²Ф=1)
=> secФcosecФ+1
put Ф=theta
cosec(theta)sec(theta)+1