Question

If A = cos^2θ + sin^4θ, then prove that for all values of θ, 3/4≤A≤1.

Answer 1

Given,

A=cos^2θ+sin^4θ

⟹A=1−sin^2θ+sin^4θ

⟹A=1+(sin^4θ−2⋅sin^2θ⋅1/2+1/4)−1/4

⟹A=(sinθ−1/2)^2+3/4

We know,

0≤sin^2θ≤1,∀θ∈R.

∴0≤(sin2θ−1/2)2≤1/4

⟹0+3/4≤(sinθ−1/2)2+3/4≤14+3/4

∴3/4≤A≤1.(∀θ∈R)†