(1 + tan θ + sec θ) (1 + cot θ - cosec θ) (A) 0 (B) 1 (C) 2 (D) - 1

Dear student

Option (C) is correct

Justification:

(1 + tan θ + sec θ) (1 + cot θ-cosec θ)
We know that,
tan θ = sin θ/cos θ
sec θ = 1/ cos θ
cot θ = cos θ/sin θ

cosec θ = 1/sin θ
Now, substitute the above values in the given problem,
we get
= (1+ sin θ/cos θ+ 1/cos θ) (1 + cos θ/sin θ-1/sin θ)

Simplify the above equation,
= (cos θ +sin θ+1)/cos θ × (sin θ+cos θ-1)/sin θ
= (cos θ+sin θ)^2 -1^2/(cos θ sin θ)
= (cos^2 θ + sin^2 θ + 2cos θ sin θ-1)/(cos θ sin θ)
= (1+ 2cos θ sin θ-1)/(cos θ sin θ)
(Since cos^2 θ + sin^2 θ = 1)

= (2cos θ sin θ)/(cos θ sin θ)
= 2
Therefore, (1 + tan θ + sec θ) (1 + cot θ-cosec θ) =2

Thanks