In a book called Introduction to tensor analysis and the calculus of moving surfaces equation (2.6) gives a formula to calculate the dot product between two vectors in terms of length alone. That formula is the following:
$$ \mathbf{U}\cdot{V} = \frac{\left| \mathbf{U} + \mathbf{V} \right|^2 - \left| \mathbf{U} - \mathbf{V} \right|^2}{4} $$
I suppose then that the above shall be equal to
$$ \mathbf{U}\cdot\mathbf{V} = |\mathbf{U}||\mathbf{V}|\cos{\alpha} $$
where $\alpha$ is the angle between $\mathbf{U}$ and $\mathbf{V}$. How can it be proved that both identities are equal?