2021 · Q50
What is the maximum value of 1 + 2sin²θcos²θ − sin⁴θ − cos⁴θ where 0° < θ < 90° ?
Elementary MathematicsTrigonometry
- A1
- B2
- C3
- D4
Explanation
Since sin⁴θ + cos⁴θ = (sin²θ + cos²θ)² − 2sin²θcos²θ = 1 − 2sin²θcos²θ, the expression becomes 1 + 2sin²θcos²θ − (1 − 2sin²θcos²θ) = 4sin²θcos²θ = sin²2θ. Its maximum value is 1, option (a).