Choose the correct option. Justify your choice. (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =?. (A) 0 (B) 1 (C) 2 (D) – 1

Answer: (C) 2

(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 – 12/(cos θ sin θ)

= (cos2θ + sin2θ + 2cos θ × sin θ -1)/(cos θ sin θ)

= (1+ 2cos θ × sin θ -1)/(cos θ × sin θ)

Since cos2θ + sin2θ = 1

= (2cos θ sin θ)/(cos θ sin θ) = 2

Therefore, (1 + tan θ + sec θ) (1 + cot θ – cosec θ) = 2

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