(1+ tanθ + secθ)(1+cotθ – cosecθ)/(secθ + tanθ)(1-sinθ) is equal to:
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. 2secθ.
Let's solve the numerator part first,
Numerator :
⇒ (1 + tanθ + secθ)(1 + cotθ - cosecθ)
⇒ (1 + sinθ/cosθ + 1/cosθ)(1 + cosθ/sinθ - 1/sinθ)
⇒ (cosθ + sinθ + 1)/cosθ × (sinθ + cosθ - 1)/sinθ
⇒ [(cosθ + sinθ)2 - 12]/sinθcosθ
⇒ [cos2θ + sin2θ + 2sinθcosθ - 1]/sinθcosθ
⇒ 2
Denominator :
⇒ (secθ + tanθ)(1 - sinθ)
⇒ (1/cosθ + sinθ/cosθ)(1 - sinθ)
⇒ [(1 + sinθ)/cosθ](1 - sinθ)
⇒ (1 - sin2θ)/cosθ
⇒ (cosθ)
Final answer = 2/cosθ
= 2secθ