If A = cosθ^2 + sinθ^4 for all values of θ, then prove that

Question

If A = cos2θ + sin4θ for all values of θ, then prove that 3/4≤A≤1

Answer 1

We have A =cos2θ + sin4θ = cos2θ + sin2θ sin2θ ≤ cos2θ + sin2θ

Therefore, A ≤ 1

Also, A = cos2θ + sin4θ = (1 – sin2θ) + sin4θ

Hence, 3/4≤A≤1