The value of (1 + tan θ + sec θ)(1 + cot θ − cosec θ) is:

1. −4

2. −1

3. 1

4. 2

Given,

(1 + tan θ + sec θ)(1 + cot θ − cosec θ)

$\Rightarrow \Big(1 + \dfrac{\sin \theta}{\cos \theta} + \dfrac{1}{\cos \theta}\Big) \Big(1 + \dfrac{\cos \theta}{\sin \theta} - \dfrac{1}{\sin \theta}\Big)\\[1em] \Rightarrow \Big( \dfrac{\cos \theta + \sin \theta + 1}{\cos \theta} \Big) \Big(\dfrac{\sin \theta + \cos \theta - 1}{\sin \theta} \Big)\\[1em] \Rightarrow \dfrac{(\cos \theta + \sin \theta)^2 - 1^2}{\cos \theta \sin \theta} \\[1em] \Rightarrow \dfrac{\cos^2 \theta + \sin^2 \theta + 2\sin \theta \cos \theta - 1}{\cos \theta \sin \theta} \\[1em] \Rightarrow \dfrac{1 + 2\sin \theta \cos \theta - 1}{\cos \theta \sin \theta} \\[1em] \Rightarrow \dfrac{ 2\sin \theta \cos \theta}{\cos \theta \sin \theta} \\[1em] \Rightarrow 2.$

Hence, option 4 is the correct option.