The equation cos 4 ⁡ x - sin 4 ⁡ x + cos ⁡ 2 x + α 2 + α = 0 will have at least one solution, if

Step-by-step solution

cos 2 ⁡ x - sin 2 ⁡ x cos 2 ⁡ x + sin 2 ⁡ x + cos ⁡ 2 x + α 2 + α = 0 ⇒ cos ⁡ 2 x × 1 + cos ⁡ 2 x + α 2 + α = 0 ⇒ α 2 + α = - 2 cos ⁡ 2 x RHS ∈ - 2,2 for a solution, - 2 ≤ α 2 + α ≤ 2 α 2 + α + 2 ≥ 0 and α 2 + α - 2 ≤ 0 Now, α 2 + α + 2 ≥ 0 is always true and α + 2 α - 1 ≤ 0 ⇒ α ∈ - 2,1