In a linear algebra text book, one homework question I received was:
Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$.
Where $\mathbf{a}$ and $\mathbf{b}$ are vectors in $\Bbb{R}^n$.
This is trivial to prove if we start from $\frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$ and reverse engineer it in $\Bbb{R}^2$: $$ \|\mathbf{a + b}\|^2 = a_1^2 + 2a_1b_1 + b_1^2 + a_2^2 + 2a_2b_2 + b_2^2 \\ \|\mathbf{a - b}\|^2 = a_1^2 - 2a_1b_1 + b_1^2 + a_2^2 - 2a_2b_2 + b_2^2 \\ \|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2 = 4a_1b_1 + 4a_2b_2 \\ \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2) = \frac{4}{4}(a_1b_1 + a_2b_2) \\ = a_1b_1 + a_2b_2 = \mathbf{a \cdot b} $$
But I'm worried about whether or not proofs like this are "legal", if that makes any sense. There was no wording in the question stating that I couldn't start from the right side of the identity, but I still have this strange feeling of guilt that I should've tried solving the identity starting from the left side and working in the "normal" direction.
For questions like these, is it okay to start from the right side of the identity? Would what I get out of doing the question in reverse be the same as if I did it normally?