If s i n 4 α + 4 c o s 4 β + 2 = 4 2 s i n α c o s β , α , β ∈ 0 , π , then c o s α + β - c o s α - β is equal to

Step-by-step solution

We have, sin 4 α + 4 cos 4 β + 2 = 4 2 sin   α cos β Applying AM ≥ GM , we get sin 4 α + 4 cos 4 β + 1 + 1 4 ≥ 4 s i n 4 α · c o s 4 β · 1 · 1 1 4 ⇒ sin 4 α + 4 cos 4 β + 2 ≥ 8 sin α cos β And, AM = GM ⇒ numbers are equal and positive ⇒ sin 4 α = 4 cos 4 β = 1 ∴ s i n 4 α = 1 ⇒ α = π 2 and c o s β = 1 2 ⇒ β = π 4 Hence, cos α + β - co