Correct Answer  Option 3 : 2
Formula used:
 sin θ/cos θ = tan θ

cos θ/sin θ = cot θ

sin2θ + cos2θ = 1
 cosec θ = 1/sin θ
 sec θ = 1/cos θ
 (a  b)^{2} = a^{2}  2ab + b^{2}
Calculation:
Let the required value be x.
⇒ x = (1 + cot θ  cosec θ)(1 + tan θ + sec θ)
⇒ x = (1 + cosθ/sinθ  1/sinθ)(1 + sinθ/cosθ + 1/cosθ)
⇒ x = (sinθ + cosθ  1)(cosθ + sinθ + 1)/(sinθ.cosθ)
⇒ x = [(sin θ + cosθ)^{2}  1]/(sinθ.cosθ)
⇒ x = [sin^{2}θ + cos^{2}θ + 2sinθ.cosθ  1]/(sinθ.cosθ)
∴ (1 + cot θ  cosec θ)(1 + tan θ + sec θ) = 2