2018 · Q63
If x = (sec²θ - tanθ)/(sec²θ + tanθ), then which one of the following is correct?
Elementary MathematicsTrigonometry
- A1/3 < x < 3
- Bx ∉ [1/3, 3]
- C-3 < x < -1/3
- D1/3 ≤ x ≤ 3
Explanation
Put t = tanθ (so sec²θ = 1 + t²): x = (1 + t² − t)/(1 + t² + t). Rearranged, (x − 1)t² + (x + 1)t + (x − 1) = 0 must have real t, so its discriminant (x + 1)² − 4(x − 1)² ≥ 0. This simplifies to (3x − 1)(x − 3) ≤ 0, i.e. 1/3 ≤ x ≤ 3 (both endpoints occur, at tanθ = ±1), option (D).