What is the value of [1 – tan (90° – θ) + sec (90° – θ)]/[tan (90° – θ) + sec (90° – θ) + 1]?
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Correct answer: B
Explanation
The correct answer is option 2 i.e. tan (θ/2)
[1 – tan (90° – θ) + sec (90° – θ)]/[tan (90° – θ) + sec (90° – θ) + 1]
= [(1 – cotθ + cosecθ)/(cotθ + cosecθ + 1)]
= [(sinθ - cosθ + 1)/sinθ]/[(cosθ + 1 + sinθ)/sinθ] = (sinθ - cosθ + 1)/(sinθ + cosθ + 1)
= (2sin θ/2 cos θ/2 + 2sin2 θ/2)/(2sin θ/2 cos θ/2 + 2cos2 θ/2)
= [2sin θ/2(cos θ/2 + sin θ/2)]/[2cos θ/2(sin θ/2 + cos θ/2)]
= tan θ/2