Find the value of the following expression.
(1 + tan θ + sec θ ) × (1 + cot θ − cosec θ)
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e. 2.
Given that, (1 + tan θ + sec θ ) × (1 + cot θ − cosec θ)
tan θ = sin θ/cos θ
cot θ = cos θ/sin θ
sec θ = 1/cos θ
cosec θ = 1/sin θ
Also, sin2 θ + cos2 θ = 1
Putting these values in the given expression,
⇒ (1 + sin θ/cos θ + 1/cos θ) × (1 + cos θ/sin θ - 1/sin θ)
⇒ (cos θ + sin θ + 1)/cos θ × (sin θ + cos θ - 1)/sin θ
⇒ (cos θ + sin θ + 1) × (sin θ + cos θ - 1)/sin θ cos θ
⇒ (cos θ sin θ + sin2 θ + sin θ + cos2 θ + cos θ sin θ + cos θ - cos θ - sin θ - 1)/sin θ cos θ
⇒ (sin2 θ + cos2 θ + 2 cos θ sin θ - 1)/cos θ sin θ
⇒ (1 + 2 cos θ sin θ - 1)/cos θ sin θ
⇒ 2 cos θ sin θ/cos θ sin θ = 2
Hence the value of the given expression is 2.