For θ ∈ [ 0 , π ] , let f ( θ ) = sin ( cos θ ) and g ( θ ) = cos ( sin θ ) . Let a = max 0 ≤ θ ≤ π f θ , b = min 0 ≤ θ = π f θ , c = max 0 ≤ 9 ≤ π g θ and d = min 0 < θ < π g θ . The correct inequalities satisfied by a ,   b ,   c ,   d are

Step-by-step solution

f ( θ ) = sin ( cos θ ) g ( θ ) = cos ( sin θ ) f ' ( θ ) = cos ( cos θ ) ( - sin θ ) < 0 ∀ θ ∈ [ 0 , π ] ∴ f ( θ ) decreases monotonically ∴ a = max f ( θ ) = f ( 0 ) = sin 1 b = min f ( θ ) = f ( π ) = - sin 1 g ' ( θ ) = - sin ( sin θ ) cos θ g ( θ ) = 1 ; g ( π ) = 1 ; g π 2 = cos 1 ∴ c = max g ( θ ) = 1 d = min g ( θ ) = cos 1 ∴ b < d < a < c