If 11 sin2 θ - cos2 θ + 4 sin θ - 4 = 0, 0∘ < θ < 90∘, then what is the value of (cos 2θ + cot 2 θ)/(sec 2 θ - tan 2 θ)
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e (12+7√{3})/(6)
Explanation:
11 sin2 θ - cos2 θ + 4 sin θ - 4 = 0
11 sin2 θ - (1 - sin2 θ) + 4 sin θ - 4 = 0
12 sin2 θ + 4 sin θ - 5 = 0
12 sin2 θ + 10 sin θ - 6sin θ - 5 = 0
2 sin θ(6 sinθ + 5) -1(6 sinθ + 5) = 0
(2sin θ - 1)(6 sinθ + 5) = 0
For 0o < θ < 90o,
sin θ = 1/2
θ = 30o
(cos 2θ + cot 2 θ)/(sec 2 θ - tan 2 θ)
On putting the value of θ,
(cos 2 × 30 + cot 2 × 30)/(sec 2 × 30 - tan 2 × 30)
(cos 60 + cot 60)/(sec60 - tan 60)
(1/2 + 1/√3)/(2 - √3)
(2 + √3)/(2 √3(2 - √3))
(2 + √3)/(4√3- 6)
(2 + √3)/(4√3- 6) × (4√3 + 6)/(4√3 + 6)
((2 + √3)(4√3 + 6))/((4√3)2- 62)
(8√3 + 12 + 12 + 6√3)/(12)
(12 + 7√3)/(6)