= Orthogonality of trigonometric functions
{title2=$\int_0^L\sin(n\pi x/L)\sin(m\pi x/L)\,dx=(L/2)\delta_{mn}$}
For positive integers $m,n$, the <sine> functions form an <orthogonal> family on $(0,L)$:
$$
\int_0^L\sin\left(\frac{n\pi x}{L}\right)\sin\left(\frac{m\pi x}{L}\right)\,dx=\frac L2\delta_{mn}.
$$
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 $L/2$.
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