2017 · Q52
Consider the following statements :
- 1If 45° < θ < 60°, then sec²θ + cosec²θ = α² for some real number α > 1. 2. If 0° < θ < 45°, then (1 + cos θ)/(1 − cos θ) = x² for some real number x > 2. 3. If 0° < θ < 45°, then cos θ/(1 − tan θ) + sin θ/(1 − cot θ) ≥ 2. What is the number of true statements ?
Elementary MathematicsTrigonometry
- AZero
- BOne
- CTwo
- DThree
Explanation
Statement 1: sec²θ+cosec²θ = 4/sin²2θ; for 45°<θ<60°, sin²2θ ∈ (3/4,1], so the sum ∈ [4,16/3), which is α² with α ≥ 2 > 1 — true. Statement 2: (1+cosθ)/(1−cosθ) = cot²(θ/2); for 0°<θ<45°, cot(θ/2) > cot 22.5° = √2+1 ≈ 2.41 > 2 — true. Statement 3: cosθ/(1−tanθ)+sinθ/(1−cotθ) = cosθ+sinθ ≤ √2 < 2 on (0°,45°) — false. Two statements are true. Hence option (C) Two.