 Orthogonality of trigonometric functions

For positive integers $m,n$, the sine functions form an orthogonal family on $(0,L)$:The same formula holds for the corresponding cosine functions when both indices are positive. Product-to-sum identities reduce the distinct-index products to cosines with nonzero integer frequencies, each with zero integral; for matching indices, the squared sine or cosine has mean $1/2$. This proves the formula and explains why separate normal modes of a uniform fixed-end string make additive contributions to its kinetic energy and potential energy. The constant cosine has squared norm $L$ rather than $L/2$.