Explanation:

cotθ / (1-sinθ)(secθ+tanθ)

= cot θ(sec²θ - tan²θ) / ( 1- sinθ)( secθ + tanθ)
= cotθ(secθ - tanθ) ( secθ +tanθ) / (1- sinθ) (secθ+tanθ)
= cosθ/sinθ (1/cosθ-sinθ/cosθ) / (1-sinθ)
1/sinθ -1 / (1-sinθ)
1- sinθ / sinθ x 1 / 1-sinθ
= 1/sinθ = cosecθ

Hence option 3 is the correct answer.