 Solution

For nonzero vectors with angle $\theta$, $|\mathbf{a}\times \mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin \theta$ and $\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta$. Hence ${\sin }^2\theta =1-{\cos }^2\theta$ gives Lagrange's identityIf either vector is zero both sides vanish, so this covers every case. Alternatively, expansion of the three cross product components gives the same identity without introducing an angle.