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

\[F(k_1) E(k_2) + F(k_2) E(k_1) - F(k_1) F(k_2) = \frac{\pi}{2}\]

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 \(0\). \(\blacksquare\)