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 = a2 - 2ab + b2
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 = [sin2θ + cos2θ + 2sinθ.cosθ - 1]/(sinθ.cosθ)
∴ (1 + cot θ - cosec θ)(1 + tan θ + sec θ) = 2
