We show that if X and Y are two non-zero Hilbert–Schmidt operators, then for any λ ≥ 0, cos2 ΘX,Y ≤ 1 1 + λ √ cosΘ|X∗ |,|Y∗ | √ cosΘ|X|,|Y| |⟨X,Y⟩| ∥X∥ 2 ∥Y∥ 2 + λ 1 + λ cosΘ|X∗ |,|Y∗ | cosΘ|X|,|Y| ≤ cosΘ|X∗ |,|Y∗ | cosΘ|X|,|Y| . Here ΘA,B denotes the angle between non-zero Hilbert–Schmidt operators A and B. This enables us to present some inequalities for the Hilbert–Schmidt norm. In particular, we prove that ∥∥∥X + Y∥∥∥ 2 ≤ √ √ 2 + 1 2 ∥∥∥ |X| + |Y| ∥∥∥ 2