**Proposition 2.2.** (Legendre's Identity). *Suppose \(k_1^2 + k_2^2 = 1\). Then*

*holds.*

*Proof.* We leave the details of this proof to the reader. Taking the derivative with respect to \(k_1\) shows that the L.H.S. is constant. To see that this value is \(\pi/2\), we take the limit as \(k_1\) goes to \(0\). \(\blacksquare\)